| L(s) = 1 | − 10·7-s + 7·9-s + 6·17-s + 4·23-s + 15·25-s + 44·31-s − 20·41-s + 10·47-s + 47·49-s − 70·63-s + 10·71-s − 8·73-s + 64·79-s + 37·81-s + 4·89-s − 8·103-s + 36·113-s − 60·119-s + 42·121-s + ⋯ |
| L(s) = 1 | − 3.77·7-s + 7/3·9-s + 1.45·17-s + 0.834·23-s + 3·25-s + 7.90·31-s − 3.12·41-s + 1.45·47-s + 47/7·49-s − 8.81·63-s + 1.18·71-s − 0.936·73-s + 7.20·79-s + 37/9·81-s + 0.423·89-s − 0.788·103-s + 3.38·113-s − 5.50·119-s + 3.81·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.028970601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.028970601\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( ( 1 + T^{2} )^{5} \) |
| good | 3 | \( 1 - 7 T^{2} + 4 p T^{4} - T^{6} + 191 T^{8} - 1112 T^{10} + 191 p^{2} T^{12} - p^{4} T^{14} + 4 p^{7} T^{16} - 7 p^{8} T^{18} + p^{10} T^{20} \) |
| 5 | \( 1 - 3 p T^{2} + 72 T^{4} - 141 T^{6} + 1211 T^{8} - 10544 T^{10} + 1211 p^{2} T^{12} - 141 p^{4} T^{14} + 72 p^{6} T^{16} - 3 p^{9} T^{18} + p^{10} T^{20} \) |
| 7 | \( ( 1 + 5 T + 2 p T^{2} + 13 T^{3} + 23 T^{4} + 24 T^{5} + 23 p T^{6} + 13 p^{2} T^{7} + 2 p^{4} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 11 | \( 1 - 42 T^{2} + 933 T^{4} - 1192 p T^{6} + 140754 T^{8} - 1433852 T^{10} + 140754 p^{2} T^{12} - 1192 p^{5} T^{14} + 933 p^{6} T^{16} - 42 p^{8} T^{18} + p^{10} T^{20} \) |
| 17 | \( ( 1 - 3 T + 44 T^{2} - 129 T^{3} + 1115 T^{4} - 3256 T^{5} + 1115 p T^{6} - 129 p^{2} T^{7} + 44 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 19 | \( 1 - 54 T^{2} + 773 T^{4} - 1960 T^{6} + 355298 T^{8} - 13566724 T^{10} + 355298 p^{2} T^{12} - 1960 p^{4} T^{14} + 773 p^{6} T^{16} - 54 p^{8} T^{18} + p^{10} T^{20} \) |
| 23 | \( ( 1 - 2 T + 35 T^{2} - 88 T^{3} + 794 T^{4} - 908 T^{5} + 794 p T^{6} - 88 p^{2} T^{7} + 35 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 29 | \( 1 - 98 T^{2} + 4277 T^{4} - 121496 T^{6} + 2369170 T^{8} - 44070220 T^{10} + 2369170 p^{2} T^{12} - 121496 p^{4} T^{14} + 4277 p^{6} T^{16} - 98 p^{8} T^{18} + p^{10} T^{20} \) |
| 31 | \( ( 1 - 22 T + 291 T^{2} - 2784 T^{3} + 21218 T^{4} - 130836 T^{5} + 21218 p T^{6} - 2784 p^{2} T^{7} + 291 p^{3} T^{8} - 22 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 37 | \( 1 - 183 T^{2} + 16504 T^{4} - 996949 T^{6} + 46509579 T^{8} - 1835893072 T^{10} + 46509579 p^{2} T^{12} - 996949 p^{4} T^{14} + 16504 p^{6} T^{16} - 183 p^{8} T^{18} + p^{10} T^{20} \) |
| 41 | \( ( 1 + 10 T + 169 T^{2} + 1280 T^{3} + 12894 T^{4} + 72364 T^{5} + 12894 p T^{6} + 1280 p^{2} T^{7} + 169 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 43 | \( 1 - 279 T^{2} + 36764 T^{4} - 3112113 T^{6} + 192982383 T^{8} - 9320199224 T^{10} + 192982383 p^{2} T^{12} - 3112113 p^{4} T^{14} + 36764 p^{6} T^{16} - 279 p^{8} T^{18} + p^{10} T^{20} \) |
| 47 | \( ( 1 - 5 T + 58 T^{2} - T^{3} + 99 T^{4} + 22600 T^{5} + 99 p T^{6} - p^{2} T^{7} + 58 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 53 | \( 1 - 358 T^{2} + 63381 T^{4} - 7246376 T^{6} + 591857634 T^{8} - 36076854564 T^{10} + 591857634 p^{2} T^{12} - 7246376 p^{4} T^{14} + 63381 p^{6} T^{16} - 358 p^{8} T^{18} + p^{10} T^{20} \) |
| 59 | \( 1 - 522 T^{2} + 126021 T^{4} - 18525368 T^{6} + 1836317778 T^{8} - 128255277884 T^{10} + 1836317778 p^{2} T^{12} - 18525368 p^{4} T^{14} + 126021 p^{6} T^{16} - 522 p^{8} T^{18} + p^{10} T^{20} \) |
| 61 | \( 1 - 246 T^{2} + 27109 T^{4} - 1780328 T^{6} + 85891362 T^{8} - 4385498372 T^{10} + 85891362 p^{2} T^{12} - 1780328 p^{4} T^{14} + 27109 p^{6} T^{16} - 246 p^{8} T^{18} + p^{10} T^{20} \) |
| 67 | \( 1 - 310 T^{2} + 51109 T^{4} - 5785768 T^{6} + 506940258 T^{8} - 36882551300 T^{10} + 506940258 p^{2} T^{12} - 5785768 p^{4} T^{14} + 51109 p^{6} T^{16} - 310 p^{8} T^{18} + p^{10} T^{20} \) |
| 71 | \( ( 1 - 5 T + 2 p T^{2} - 1245 T^{3} + 15703 T^{4} - 100696 T^{5} + 15703 p T^{6} - 1245 p^{2} T^{7} + 2 p^{4} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 73 | \( ( 1 + 4 T + 93 T^{2} - 160 T^{3} - 3030 T^{4} - 56904 T^{5} - 3030 p T^{6} - 160 p^{2} T^{7} + 93 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 79 | \( ( 1 - 32 T + 683 T^{2} - 10432 T^{3} + 127594 T^{4} - 1250880 T^{5} + 127594 p T^{6} - 10432 p^{2} T^{7} + 683 p^{3} T^{8} - 32 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 83 | \( 1 - 370 T^{2} + 68325 T^{4} - 8153976 T^{6} + 739677314 T^{8} - 61066134636 T^{10} + 739677314 p^{2} T^{12} - 8153976 p^{4} T^{14} + 68325 p^{6} T^{16} - 370 p^{8} T^{18} + p^{10} T^{20} \) |
| 89 | \( ( 1 - 2 T + 309 T^{2} - 632 T^{3} + 47266 T^{4} - 77516 T^{5} + 47266 p T^{6} - 632 p^{2} T^{7} + 309 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 97 | \( ( 1 + 325 T^{2} + 752 T^{3} + 47258 T^{4} + 145184 T^{5} + 47258 p T^{6} + 752 p^{2} T^{7} + 325 p^{3} T^{8} + p^{5} T^{10} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.02586266114241951971117475008, −2.91145212789052205191288207586, −2.90689153183871802958036460083, −2.89262583765992781161820552014, −2.82451208954641793887258234717, −2.45666883964982631957411232088, −2.33076714037990729847190071929, −2.28828055602796118317971124724, −2.27285277076652398211647787554, −2.16698898230070645161769671204, −1.98738259066472749569406486465, −1.96928012324912441347537904161, −1.93354509112435761981041864430, −1.66444780016980621372359193610, −1.49352531766003419573540796955, −1.14428563266913065714111824810, −1.12678110508293559624363244650, −0.953890543445829871237657063031, −0.946753958689714618805258522391, −0.942925172228745075400159966096, −0.913881356778431432627693846761, −0.870895587628678116902094873383, −0.55927554402220469330941762884, −0.37299904123041270897657546820, −0.04307868830021876206668877779,
0.04307868830021876206668877779, 0.37299904123041270897657546820, 0.55927554402220469330941762884, 0.870895587628678116902094873383, 0.913881356778431432627693846761, 0.942925172228745075400159966096, 0.946753958689714618805258522391, 0.953890543445829871237657063031, 1.12678110508293559624363244650, 1.14428563266913065714111824810, 1.49352531766003419573540796955, 1.66444780016980621372359193610, 1.93354509112435761981041864430, 1.96928012324912441347537904161, 1.98738259066472749569406486465, 2.16698898230070645161769671204, 2.27285277076652398211647787554, 2.28828055602796118317971124724, 2.33076714037990729847190071929, 2.45666883964982631957411232088, 2.82451208954641793887258234717, 2.89262583765992781161820552014, 2.90689153183871802958036460083, 2.91145212789052205191288207586, 3.02586266114241951971117475008
Plot not available for L-functions of degree greater than 10.
