| L(s) = 1 | − 5.77e9·2-s + 5.55e15·3-s − 1.14e20·4-s + 1.87e23·5-s − 3.21e25·6-s + 3.73e28·7-s + 1.51e30·8-s + 3.09e31·9-s − 1.08e33·10-s − 1.19e35·11-s − 6.34e35·12-s − 2.41e37·13-s − 2.15e38·14-s + 1.04e39·15-s + 8.12e39·16-s + 2.68e40·17-s − 1.78e41·18-s − 8.49e42·19-s − 2.14e43·20-s + 2.07e44·21-s + 6.87e44·22-s + 2.56e45·23-s + 8.40e45·24-s − 3.25e46·25-s + 1.39e47·26-s + 1.71e47·27-s − 4.26e48·28-s + ⋯ |
| L(s) = 1 | − 0.475·2-s + 0.577·3-s − 0.773·4-s + 0.720·5-s − 0.274·6-s + 1.82·7-s + 0.843·8-s + 0.333·9-s − 0.342·10-s − 1.54·11-s − 0.446·12-s − 1.16·13-s − 0.868·14-s + 0.416·15-s + 0.373·16-s + 0.161·17-s − 0.158·18-s − 1.23·19-s − 0.557·20-s + 1.05·21-s + 0.735·22-s + 0.618·23-s + 0.486·24-s − 0.480·25-s + 0.552·26-s + 0.192·27-s − 1.41·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(68-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+67/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(34)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{69}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 5.55e15T \) |
| good | 2 | \( 1 + 5.77e9T + 1.47e20T^{2} \) |
| 5 | \( 1 - 1.87e23T + 6.77e46T^{2} \) |
| 7 | \( 1 - 3.73e28T + 4.18e56T^{2} \) |
| 11 | \( 1 + 1.19e35T + 5.93e69T^{2} \) |
| 13 | \( 1 + 2.41e37T + 4.30e74T^{2} \) |
| 17 | \( 1 - 2.68e40T + 2.75e82T^{2} \) |
| 19 | \( 1 + 8.49e42T + 4.74e85T^{2} \) |
| 23 | \( 1 - 2.56e45T + 1.72e91T^{2} \) |
| 29 | \( 1 + 8.53e48T + 9.56e97T^{2} \) |
| 31 | \( 1 - 9.91e49T + 8.34e99T^{2} \) |
| 37 | \( 1 + 1.57e52T + 1.17e105T^{2} \) |
| 41 | \( 1 - 1.78e54T + 1.13e108T^{2} \) |
| 43 | \( 1 + 5.13e54T + 2.76e109T^{2} \) |
| 47 | \( 1 + 1.02e56T + 1.07e112T^{2} \) |
| 53 | \( 1 + 9.83e57T + 3.36e115T^{2} \) |
| 59 | \( 1 + 2.18e59T + 4.43e118T^{2} \) |
| 61 | \( 1 - 5.16e59T + 4.14e119T^{2} \) |
| 67 | \( 1 - 8.46e60T + 2.22e122T^{2} \) |
| 71 | \( 1 - 2.04e61T + 1.08e124T^{2} \) |
| 73 | \( 1 + 1.75e62T + 6.96e124T^{2} \) |
| 79 | \( 1 + 1.13e62T + 1.38e127T^{2} \) |
| 83 | \( 1 - 1.01e64T + 3.78e128T^{2} \) |
| 89 | \( 1 + 2.83e65T + 4.06e130T^{2} \) |
| 97 | \( 1 + 4.28e66T + 1.29e133T^{2} \) |
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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
−12.88136108571970466307866441639, −10.81435304953231774461706165710, −9.715428836116431961095148489360, −8.335544692364767818594999996779, −7.67143046955951599675852070128, −5.26067610284870270969572141356, −4.52948165964913068968771851212, −2.44204322781610164922381550489, −1.53828411247393345389190960036, 0,
1.53828411247393345389190960036, 2.44204322781610164922381550489, 4.52948165964913068968771851212, 5.26067610284870270969572141356, 7.67143046955951599675852070128, 8.335544692364767818594999996779, 9.715428836116431961095148489360, 10.81435304953231774461706165710, 12.88136108571970466307866441639
