| L(s) = 1 | − 14·4-s − 7·9-s + 95·16-s + 24·19-s − 52·25-s + 98·36-s + 24·37-s + 46·49-s − 422·64-s + 32·67-s + 4·73-s − 336·76-s + 17·81-s + 728·100-s + 84·103-s − 116·121-s + 127-s + 131-s + 137-s + 139-s − 665·144-s − 336·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 7·4-s − 7/3·9-s + 95/4·16-s + 5.50·19-s − 10.3·25-s + 49/3·36-s + 3.94·37-s + 46/7·49-s − 52.7·64-s + 3.90·67-s + 0.468·73-s − 38.5·76-s + 17/9·81-s + 72.7·100-s + 8.27·103-s − 10.5·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 55.4·144-s − 27.6·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 67^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 67^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1410436648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1410436648\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 7 T^{2} + 32 T^{4} + 128 T^{6} + 47 p^{2} T^{8} + 1186 T^{10} + 47 p^{4} T^{12} + 128 p^{4} T^{14} + 32 p^{6} T^{16} + 7 p^{8} T^{18} + p^{10} T^{20} \) |
| 67 | \( ( 1 - 16 T + 95 T^{2} - 280 T^{3} - 2006 T^{4} + 39184 T^{5} - 2006 p T^{6} - 280 p^{2} T^{7} + 95 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| good | 2 | \( ( 1 + 7 T^{2} + 13 p T^{4} + 71 T^{6} + 173 T^{8} + 93 p^{2} T^{10} + 173 p^{2} T^{12} + 71 p^{4} T^{14} + 13 p^{7} T^{16} + 7 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 5 | \( ( 1 + 26 T^{2} + 367 T^{4} + 3506 T^{6} + 25161 T^{8} + 141088 T^{10} + 25161 p^{2} T^{12} + 3506 p^{4} T^{14} + 367 p^{6} T^{16} + 26 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 7 | \( ( 1 - 23 T^{2} + 283 T^{4} - 2507 T^{6} + 19167 T^{8} - 139372 T^{10} + 19167 p^{2} T^{12} - 2507 p^{4} T^{14} + 283 p^{6} T^{16} - 23 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 11 | \( ( 1 + 58 T^{2} + 1862 T^{4} + 40628 T^{6} + 660125 T^{8} + 8230836 T^{10} + 660125 p^{2} T^{12} + 40628 p^{4} T^{14} + 1862 p^{6} T^{16} + 58 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 13 | \( ( 1 - 57 T^{2} + 1668 T^{4} - 35816 T^{6} + 48091 p T^{8} - 8962430 T^{10} + 48091 p^{3} T^{12} - 35816 p^{4} T^{14} + 1668 p^{6} T^{16} - 57 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 17 | \( ( 1 - 69 T^{2} + 2897 T^{4} - 84448 T^{6} + 1943438 T^{8} - 36126838 T^{10} + 1943438 p^{2} T^{12} - 84448 p^{4} T^{14} + 2897 p^{6} T^{16} - 69 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 19 | \( ( 1 - 6 T + 77 T^{2} - 21 p T^{3} + 2694 T^{4} - 10816 T^{5} + 2694 p T^{6} - 21 p^{3} T^{7} + 77 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} )^{4} \) |
| 23 | \( ( 1 - 135 T^{2} + 9753 T^{4} - 466864 T^{6} + 16291345 T^{8} - 428543479 T^{10} + 16291345 p^{2} T^{12} - 466864 p^{4} T^{14} + 9753 p^{6} T^{16} - 135 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 29 | \( ( 1 - 155 T^{2} + 11412 T^{4} - 541365 T^{6} + 19456967 T^{8} - 596118168 T^{10} + 19456967 p^{2} T^{12} - 541365 p^{4} T^{14} + 11412 p^{6} T^{16} - 155 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 31 | \( ( 1 - 128 T^{2} + 9279 T^{4} - 470100 T^{6} + 18750365 T^{8} - 628047028 T^{10} + 18750365 p^{2} T^{12} - 470100 p^{4} T^{14} + 9279 p^{6} T^{16} - 128 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 37 | \( ( 1 - 6 T + 126 T^{2} - 349 T^{3} + 5773 T^{4} - 8281 T^{5} + 5773 p T^{6} - 349 p^{2} T^{7} + 126 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} )^{4} \) |
| 41 | \( ( 1 + 273 T^{2} + 36843 T^{4} + 3219131 T^{6} + 201840205 T^{8} + 231534202 p T^{10} + 201840205 p^{2} T^{12} + 3219131 p^{4} T^{14} + 36843 p^{6} T^{16} + 273 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 43 | \( ( 1 - 193 T^{2} + 18277 T^{4} - 1220287 T^{6} + 67258147 T^{8} - 3150400198 T^{10} + 67258147 p^{2} T^{12} - 1220287 p^{4} T^{14} + 18277 p^{6} T^{16} - 193 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 47 | \( ( 1 - 168 T^{2} + 14919 T^{4} - 890619 T^{6} + 44275280 T^{8} - 2059302358 T^{10} + 44275280 p^{2} T^{12} - 890619 p^{4} T^{14} + 14919 p^{6} T^{16} - 168 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 53 | \( ( 1 + 308 T^{2} + 47691 T^{4} + 4933406 T^{6} + 378892457 T^{8} + 22619448350 T^{10} + 378892457 p^{2} T^{12} + 4933406 p^{4} T^{14} + 47691 p^{6} T^{16} + 308 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 59 | \( ( 1 - 125 T^{2} + 15350 T^{4} - 875993 T^{6} + 60034785 T^{8} - 2515766660 T^{10} + 60034785 p^{2} T^{12} - 875993 p^{4} T^{14} + 15350 p^{6} T^{16} - 125 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 61 | \( ( 1 - 295 T^{2} + 47942 T^{4} - 5519398 T^{6} + 481865517 T^{8} - 32973713998 T^{10} + 481865517 p^{2} T^{12} - 5519398 p^{4} T^{14} + 47942 p^{6} T^{16} - 295 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 71 | \( ( 1 - 460 T^{2} + 100726 T^{4} - 14076066 T^{6} + 1429238317 T^{8} - 113368312844 T^{10} + 1429238317 p^{2} T^{12} - 14076066 p^{4} T^{14} + 100726 p^{6} T^{16} - 460 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 73 | \( ( 1 - T + 206 T^{2} - 103 T^{3} + 20115 T^{4} - 6904 T^{5} + 20115 p T^{6} - 103 p^{2} T^{7} + 206 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{4} \) |
| 79 | \( ( 1 - 411 T^{2} + 63834 T^{4} - 3197614 T^{6} - 304842923 T^{8} + 50995950578 T^{10} - 304842923 p^{2} T^{12} - 3197614 p^{4} T^{14} + 63834 p^{6} T^{16} - 411 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 83 | \( ( 1 - 365 T^{2} + 75403 T^{4} - 11235039 T^{6} + 1291616557 T^{8} - 118995779466 T^{10} + 1291616557 p^{2} T^{12} - 11235039 p^{4} T^{14} + 75403 p^{6} T^{16} - 365 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 89 | \( ( 1 - 703 T^{2} + 234596 T^{4} - 48864245 T^{6} + 7035486623 T^{8} - 731920278252 T^{10} + 7035486623 p^{2} T^{12} - 48864245 p^{4} T^{14} + 234596 p^{6} T^{16} - 703 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 97 | \( ( 1 - 684 T^{2} + 228862 T^{4} - 49086542 T^{6} + 7453302405 T^{8} - 836001815156 T^{10} + 7453302405 p^{2} T^{12} - 49086542 p^{4} T^{14} + 228862 p^{6} T^{16} - 684 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.29459152706276082895232039990, −3.29206770561001190995062681470, −3.23840874400094610942362383184, −3.07826075359553963764863037214, −2.69918336779401678554668152290, −2.67410437813553966203515815590, −2.66233779975090210601617870606, −2.64627075109504740437487492197, −2.48920001476173565348504260440, −2.48426108148139774040314413064, −2.46007016053370844431733058977, −2.30130994792647939470563859693, −2.22954449860321325758363458902, −2.00245856161857353542131417613, −1.86415770901429275964449738495, −1.78504750963046853897402922742, −1.56290759062677768781585142089, −1.51468012898651421800027972857, −1.40723035470048483597588389237, −1.14771223180649879171000675561, −0.896311341376370666524894428281, −0.66211561548302443978379683474, −0.65690523542256897417039213024, −0.50186582770773769812744707004, −0.29122689763761058850434738369,
0.29122689763761058850434738369, 0.50186582770773769812744707004, 0.65690523542256897417039213024, 0.66211561548302443978379683474, 0.896311341376370666524894428281, 1.14771223180649879171000675561, 1.40723035470048483597588389237, 1.51468012898651421800027972857, 1.56290759062677768781585142089, 1.78504750963046853897402922742, 1.86415770901429275964449738495, 2.00245856161857353542131417613, 2.22954449860321325758363458902, 2.30130994792647939470563859693, 2.46007016053370844431733058977, 2.48426108148139774040314413064, 2.48920001476173565348504260440, 2.64627075109504740437487492197, 2.66233779975090210601617870606, 2.67410437813553966203515815590, 2.69918336779401678554668152290, 3.07826075359553963764863037214, 3.23840874400094610942362383184, 3.29206770561001190995062681470, 3.29459152706276082895232039990
Plot not available for L-functions of degree greater than 10.
