| L(s) = 1 | − 1.43e3·3-s + 4.09e3·4-s + 5.23e3·5-s − 2.11e5·7-s + 1.52e6·9-s − 5.86e6·12-s − 7.49e6·15-s + 1.67e7·16-s − 1.16e7·17-s − 9.38e7·19-s + 2.14e7·20-s + 3.02e8·21-s − 2.16e8·25-s − 1.41e9·27-s − 8.65e8·28-s − 6.37e8·29-s − 1.10e9·35-s + 6.23e9·36-s + 9.48e9·41-s + 7.96e9·45-s − 2.40e10·48-s + 3.07e10·49-s + 1.67e10·51-s − 7.91e9·53-s + 1.34e11·57-s + 4.21e10·59-s − 3.07e10·60-s + ⋯ |
| L(s) = 1 | − 1.96·3-s + 4-s + 0.334·5-s − 1.79·7-s + 2.86·9-s − 1.96·12-s − 0.658·15-s + 16-s − 0.483·17-s − 1.99·19-s + 0.334·20-s + 3.52·21-s − 0.887·25-s − 3.66·27-s − 1.79·28-s − 1.07·29-s − 0.601·35-s + 2.86·36-s + 1.99·41-s + 0.958·45-s − 1.96·48-s + 2.22·49-s + 0.950·51-s − 0.357·53-s + 3.92·57-s + 59-s − 0.658·60-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.6742189039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6742189039\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 - p^{6} T \) |
| good | 2 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 3 | \( 1 + 1433 T + p^{12} T^{2} \) |
| 5 | \( 1 - 5231 T + p^{12} T^{2} \) |
| 7 | \( 1 + 211273 T + p^{12} T^{2} \) |
| 11 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 13 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 17 | \( 1 + 11672638 T + p^{12} T^{2} \) |
| 19 | \( 1 + 93895513 T + p^{12} T^{2} \) |
| 23 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 29 | \( 1 + 637537633 T + p^{12} T^{2} \) |
| 31 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 37 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 41 | \( 1 - 9483946607 T + p^{12} T^{2} \) |
| 43 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 47 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 53 | \( 1 + 7914541633 T + p^{12} T^{2} \) |
| 61 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 67 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 71 | \( 1 - 210865062242 T + p^{12} T^{2} \) |
| 73 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 79 | \( 1 + 402738855433 T + p^{12} T^{2} \) |
| 83 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 89 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 97 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.60105779640211405362670502795, −11.34643054560229609511947925050, −10.53721206325074530033506681007, −9.661277348731806319473336445094, −7.14134374636648922442822990565, −6.30430258635853045707634997246, −5.83678605388361622782114589238, −4.01044719546719501778210240220, −2.09729684936529905219395962876, −0.47526207818152328414793319320,
0.47526207818152328414793319320, 2.09729684936529905219395962876, 4.01044719546719501778210240220, 5.83678605388361622782114589238, 6.30430258635853045707634997246, 7.14134374636648922442822990565, 9.661277348731806319473336445094, 10.53721206325074530033506681007, 11.34643054560229609511947925050, 12.60105779640211405362670502795
