| L(s) = 1 | + 1.52e18·2-s + 3.29e28·3-s + 1.67e36·4-s + 6.57e41·5-s + 5.03e46·6-s − 6.63e49·7-s + 1.54e54·8-s + 4.83e56·9-s + 1.00e60·10-s − 2.55e61·11-s + 5.50e64·12-s − 1.55e66·13-s − 1.01e68·14-s + 2.16e70·15-s + 1.24e72·16-s − 5.44e71·17-s + 7.39e74·18-s − 1.66e76·19-s + 1.09e78·20-s − 2.18e78·21-s − 3.90e79·22-s + 1.53e81·23-s + 5.07e82·24-s + 2.81e83·25-s − 2.38e84·26-s − 3.80e84·27-s − 1.11e86·28-s + ⋯ |
| L(s) = 1 | + 1.87·2-s + 1.34·3-s + 2.51·4-s + 1.69·5-s + 2.52·6-s − 0.345·7-s + 2.84·8-s + 0.807·9-s + 3.17·10-s − 0.278·11-s + 3.38·12-s − 0.817·13-s − 0.648·14-s + 2.27·15-s + 2.82·16-s − 0.0334·17-s + 1.51·18-s − 1.36·19-s + 4.26·20-s − 0.464·21-s − 0.522·22-s + 1.46·23-s + 3.82·24-s + 1.86·25-s − 1.53·26-s − 0.259·27-s − 0.870·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(120-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+59.5) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(60)\) |
\(\approx\) |
\(14.93516639\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.93516639\) |
| \(L(\frac{121}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 1.52e18T + 6.64e35T^{2} \) |
| 3 | \( 1 - 3.29e28T + 5.99e56T^{2} \) |
| 5 | \( 1 - 6.57e41T + 1.50e83T^{2} \) |
| 7 | \( 1 + 6.63e49T + 3.68e100T^{2} \) |
| 11 | \( 1 + 2.55e61T + 8.42e123T^{2} \) |
| 13 | \( 1 + 1.55e66T + 3.62e132T^{2} \) |
| 17 | \( 1 + 5.44e71T + 2.65e146T^{2} \) |
| 19 | \( 1 + 1.66e76T + 1.48e152T^{2} \) |
| 23 | \( 1 - 1.53e81T + 1.11e162T^{2} \) |
| 29 | \( 1 + 8.98e86T + 1.06e174T^{2} \) |
| 31 | \( 1 + 2.64e88T + 2.96e177T^{2} \) |
| 37 | \( 1 + 2.34e93T + 4.13e186T^{2} \) |
| 41 | \( 1 + 2.60e94T + 8.34e191T^{2} \) |
| 43 | \( 1 - 1.81e97T + 2.41e194T^{2} \) |
| 47 | \( 1 + 9.77e97T + 9.54e198T^{2} \) |
| 53 | \( 1 - 2.98e102T + 1.54e205T^{2} \) |
| 59 | \( 1 - 2.79e102T + 5.38e210T^{2} \) |
| 61 | \( 1 + 9.69e105T + 2.84e212T^{2} \) |
| 67 | \( 1 + 8.21e108T + 2.00e217T^{2} \) |
| 71 | \( 1 - 9.76e108T + 1.99e220T^{2} \) |
| 73 | \( 1 - 6.12e110T + 5.43e221T^{2} \) |
| 79 | \( 1 - 6.14e112T + 6.57e225T^{2} \) |
| 83 | \( 1 - 3.99e113T + 2.34e228T^{2} \) |
| 89 | \( 1 - 5.13e115T + 9.49e231T^{2} \) |
| 97 | \( 1 - 2.85e118T + 2.66e236T^{2} \) |
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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
−13.37489221886784926559451306849, −12.76254479002037209043394023473, −10.57819759027740298372703515142, −9.150388160113563621313652715167, −7.16130088965525359989800550478, −5.96544153950919642431088317344, −4.84111811684327763814102432944, −3.34895732356849513357337867446, −2.43259362262540323949526678027, −1.86262580588554411701300335351,
1.86262580588554411701300335351, 2.43259362262540323949526678027, 3.34895732356849513357337867446, 4.84111811684327763814102432944, 5.96544153950919642431088317344, 7.16130088965525359989800550478, 9.150388160113563621313652715167, 10.57819759027740298372703515142, 12.76254479002037209043394023473, 13.37489221886784926559451306849
