| L(s) = 1 | − 9·9-s − 22·11-s + 4·16-s − 2·19-s − 12·29-s + 10·31-s + 40·41-s + 15·49-s + 112·59-s + 6·61-s − 7·64-s + 42·71-s + 18·79-s + 51·81-s − 14·89-s + 198·99-s − 58·101-s + 18·109-s + 227·121-s + 127-s + 131-s + 137-s + 139-s − 36·144-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 3·9-s − 6.63·11-s + 16-s − 0.458·19-s − 2.22·29-s + 1.79·31-s + 6.24·41-s + 15/7·49-s + 14.5·59-s + 0.768·61-s − 7/8·64-s + 4.98·71-s + 2.02·79-s + 17/3·81-s − 1.48·89-s + 19.8·99-s − 5.77·101-s + 1.72·109-s + 20.6·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.439314076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.439314076\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 29 | \( ( 1 + 6 T - 13 T^{2} - 316 T^{3} - 13 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| good | 2 | \( 1 - p^{2} T^{4} + 7 T^{6} + p T^{8} - 7 T^{10} + 69 T^{12} - 7 p^{2} T^{14} + p^{5} T^{16} + 7 p^{6} T^{18} - p^{10} T^{20} + p^{12} T^{24} \) |
| 3 | \( 1 + p^{2} T^{2} + 10 p T^{4} + 14 T^{6} - 179 T^{8} - 173 p T^{10} - 1016 T^{12} - 173 p^{3} T^{14} - 179 p^{4} T^{16} + 14 p^{6} T^{18} + 10 p^{9} T^{20} + p^{12} T^{22} + p^{12} T^{24} \) |
| 7 | \( 1 - 15 T^{2} + 274 T^{4} - 2906 T^{6} + 32341 T^{8} - 257671 T^{10} + 2076880 T^{12} - 257671 p^{2} T^{14} + 32341 p^{4} T^{16} - 2906 p^{6} T^{18} + 274 p^{8} T^{20} - 15 p^{10} T^{22} + p^{12} T^{24} \) |
| 11 | \( ( 1 + p T + 68 T^{2} + 354 T^{3} + 153 p T^{4} + 6723 T^{5} + 23296 T^{6} + 6723 p T^{7} + 153 p^{3} T^{8} + 354 p^{3} T^{9} + 68 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} )^{2} \) |
| 13 | \( 1 + T^{2} + 322 T^{4} + 398 T^{6} + 82445 T^{8} + 141953 T^{10} + 16333248 T^{12} + 141953 p^{2} T^{14} + 82445 p^{4} T^{16} + 398 p^{6} T^{18} + 322 p^{8} T^{20} + p^{10} T^{22} + p^{12} T^{24} \) |
| 17 | \( ( 1 - 78 T^{2} + 2783 T^{4} - 59300 T^{6} + 2783 p^{2} T^{8} - 78 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 19 | \( ( 1 + T - 18 T^{2} - 156 T^{3} + 67 T^{4} + 1127 T^{5} + 9612 T^{6} + 1127 p T^{7} + 67 p^{2} T^{8} - 156 p^{3} T^{9} - 18 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 23 | \( 1 - 3 T^{2} + 362 T^{4} + 6234 T^{6} + 102469 T^{8} - 2383623 T^{10} + 211561280 T^{12} - 2383623 p^{2} T^{14} + 102469 p^{4} T^{16} + 6234 p^{6} T^{18} + 362 p^{8} T^{20} - 3 p^{10} T^{22} + p^{12} T^{24} \) |
| 31 | \( ( 1 - 5 T - 34 T^{2} + 388 T^{3} - 1649 T^{4} - 7283 T^{5} + 109220 T^{6} - 7283 p T^{7} - 1649 p^{2} T^{8} + 388 p^{3} T^{9} - 34 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 37 | \( 1 + p T^{2} + 2646 T^{4} + 69866 T^{6} + 2293373 T^{8} + 7815969 T^{10} + 322004152 T^{12} + 7815969 p^{2} T^{14} + 2293373 p^{4} T^{16} + 69866 p^{6} T^{18} + 2646 p^{8} T^{20} + p^{11} T^{22} + p^{12} T^{24} \) |
| 41 | \( ( 1 - 10 T + 147 T^{2} - 828 T^{3} + 147 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 43 | \( 1 + 85 T^{2} + 966 T^{4} - 40342 T^{6} + 4135613 T^{8} + 124537329 T^{10} - 4026281384 T^{12} + 124537329 p^{2} T^{14} + 4135613 p^{4} T^{16} - 40342 p^{6} T^{18} + 966 p^{8} T^{20} + 85 p^{10} T^{22} + p^{12} T^{24} \) |
| 47 | \( 1 + 85 T^{2} + 4526 T^{4} + 85162 T^{6} + 521813 T^{8} - 167792879 T^{10} - 4848096744 T^{12} - 167792879 p^{2} T^{14} + 521813 p^{4} T^{16} + 85162 p^{6} T^{18} + 4526 p^{8} T^{20} + 85 p^{10} T^{22} + p^{12} T^{24} \) |
| 53 | \( 1 + 125 T^{2} + 6390 T^{4} + 153506 T^{6} + 99357 T^{8} - 714301999 T^{10} - 61206560968 T^{12} - 714301999 p^{2} T^{14} + 99357 p^{4} T^{16} + 153506 p^{6} T^{18} + 6390 p^{8} T^{20} + 125 p^{10} T^{22} + p^{12} T^{24} \) |
| 59 | \( ( 1 - 28 T + 429 T^{2} - 4032 T^{3} + 429 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 61 | \( ( 1 - 3 T - 24 T^{2} - 74 T^{3} + 4633 T^{4} - 24883 T^{5} - 67236 T^{6} - 24883 p T^{7} + 4633 p^{2} T^{8} - 74 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 67 | \( 1 + 25 T^{2} - 3290 T^{4} + 125894 T^{6} + 42710237 T^{8} + 864472049 T^{10} - 131009765928 T^{12} + 864472049 p^{2} T^{14} + 42710237 p^{4} T^{16} + 125894 p^{6} T^{18} - 3290 p^{8} T^{20} + 25 p^{10} T^{22} + p^{12} T^{24} \) |
| 71 | \( ( 1 - 21 T + 118 T^{2} + 546 T^{3} - 4409 T^{4} - 107373 T^{5} + 1690660 T^{6} - 107373 p T^{7} - 4409 p^{2} T^{8} + 546 p^{3} T^{9} + 118 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 73 | \( 1 + 25 T^{2} - 1862 T^{4} - 47650 T^{6} - 19650187 T^{8} - 158677351 T^{10} + 197596571712 T^{12} - 158677351 p^{2} T^{14} - 19650187 p^{4} T^{16} - 47650 p^{6} T^{18} - 1862 p^{8} T^{20} + 25 p^{10} T^{22} + p^{12} T^{24} \) |
| 79 | \( ( 1 - 9 T + 2 T^{2} - 210 T^{3} - 977 T^{4} + 72339 T^{5} - 812260 T^{6} + 72339 p T^{7} - 977 p^{2} T^{8} - 210 p^{3} T^{9} + 2 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 83 | \( 1 + 17 T^{2} + 13798 T^{4} + 182630 T^{6} + 93382253 T^{8} - 3072902311 T^{10} + 610692711992 T^{12} - 3072902311 p^{2} T^{14} + 93382253 p^{4} T^{16} + 182630 p^{6} T^{18} + 13798 p^{8} T^{20} + 17 p^{10} T^{22} + p^{12} T^{24} \) |
| 89 | \( ( 1 + 7 T + 30 T^{2} - 532 T^{3} + 7389 T^{4} + 23261 T^{5} + 663944 T^{6} + 23261 p T^{7} + 7389 p^{2} T^{8} - 532 p^{3} T^{9} + 30 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 97 | \( 1 - 227 T^{2} + 55098 T^{4} - 8981686 T^{6} + 1311008117 T^{8} - 155139828183 T^{10} + 16527302027200 T^{12} - 155139828183 p^{2} T^{14} + 1311008117 p^{4} T^{16} - 8981686 p^{6} T^{18} + 55098 p^{8} T^{20} - 227 p^{10} T^{22} + p^{12} T^{24} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.60151901386523786813593172196, −3.09137976285492812063000095922, −3.01636953800108274243193360532, −2.95046832992624838034508467227, −2.85278400798487624887898166388, −2.73454311351862674479481445113, −2.68223627742258636331288972563, −2.56914338507349949829996305524, −2.45674520430436713929987062051, −2.43316760190735844865230745259, −2.37337820652177688046594700560, −2.33459634648997083888291704363, −2.29820804691159466893495465447, −2.29775728184295354990640619926, −2.14853453383160706117915099986, −2.10826222135049537453998248923, −1.53661490412266991596074230642, −1.31313743561344947150256737284, −1.25774665699899364000769370177, −1.04995192961342580902940644537, −0.78718340929108360264941588753, −0.71700945976289060022506502668, −0.59393026224928476511153861174, −0.36721865136928077191181121137, −0.29332623179668029551362981894,
0.29332623179668029551362981894, 0.36721865136928077191181121137, 0.59393026224928476511153861174, 0.71700945976289060022506502668, 0.78718340929108360264941588753, 1.04995192961342580902940644537, 1.25774665699899364000769370177, 1.31313743561344947150256737284, 1.53661490412266991596074230642, 2.10826222135049537453998248923, 2.14853453383160706117915099986, 2.29775728184295354990640619926, 2.29820804691159466893495465447, 2.33459634648997083888291704363, 2.37337820652177688046594700560, 2.43316760190735844865230745259, 2.45674520430436713929987062051, 2.56914338507349949829996305524, 2.68223627742258636331288972563, 2.73454311351862674479481445113, 2.85278400798487624887898166388, 2.95046832992624838034508467227, 3.01636953800108274243193360532, 3.09137976285492812063000095922, 3.60151901386523786813593172196
Plot not available for L-functions of degree greater than 10.
