| L(s) = 1 | − 3.54e9·2-s − 5.55e15·3-s − 1.34e20·4-s + 1.66e23·5-s + 1.97e25·6-s − 1.26e28·7-s + 1.00e30·8-s + 3.09e31·9-s − 5.90e32·10-s + 3.05e34·11-s + 7.50e35·12-s − 2.14e37·13-s + 4.50e37·14-s − 9.24e38·15-s + 1.63e40·16-s + 1.02e41·17-s − 1.09e41·18-s + 2.26e42·19-s − 2.24e43·20-s + 7.05e43·21-s − 1.08e44·22-s − 7.58e45·23-s − 5.57e45·24-s − 4.00e46·25-s + 7.61e46·26-s − 1.71e47·27-s + 1.71e48·28-s + ⋯ |
| L(s) = 1 | − 0.291·2-s − 0.577·3-s − 0.914·4-s + 0.639·5-s + 0.168·6-s − 0.620·7-s + 0.559·8-s + 0.333·9-s − 0.186·10-s + 0.396·11-s + 0.528·12-s − 1.03·13-s + 0.181·14-s − 0.368·15-s + 0.751·16-s + 0.616·17-s − 0.0973·18-s + 0.328·19-s − 0.584·20-s + 0.358·21-s − 0.115·22-s − 1.82·23-s − 0.322·24-s − 0.591·25-s + 0.302·26-s − 0.192·27-s + 0.567·28-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
| \(L(34)\) |
\(\approx\) |
\(0.7203209449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7203209449\) |
| \(L(\frac{69}{2})\) |
|
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 | \( 1 + 3.54e9T + 1.47e20T^{2} \) |
| 5 | \( 1 - 1.66e23T + 6.77e46T^{2} \) |
| 7 | \( 1 + 1.26e28T + 4.18e56T^{2} \) |
| 11 | \( 1 - 3.05e34T + 5.93e69T^{2} \) |
| 13 | \( 1 + 2.14e37T + 4.30e74T^{2} \) |
| 17 | \( 1 - 1.02e41T + 2.75e82T^{2} \) |
| 19 | \( 1 - 2.26e42T + 4.74e85T^{2} \) |
| 23 | \( 1 + 7.58e45T + 1.72e91T^{2} \) |
| 29 | \( 1 + 4.33e48T + 9.56e97T^{2} \) |
| 31 | \( 1 + 3.83e49T + 8.34e99T^{2} \) |
| 37 | \( 1 - 3.81e52T + 1.17e105T^{2} \) |
| 41 | \( 1 + 6.60e53T + 1.13e108T^{2} \) |
| 43 | \( 1 + 5.55e54T + 2.76e109T^{2} \) |
| 47 | \( 1 + 2.64e55T + 1.07e112T^{2} \) |
| 53 | \( 1 + 1.96e57T + 3.36e115T^{2} \) |
| 59 | \( 1 + 1.16e59T + 4.43e118T^{2} \) |
| 61 | \( 1 - 1.04e60T + 4.14e119T^{2} \) |
| 67 | \( 1 + 5.71e60T + 2.22e122T^{2} \) |
| 71 | \( 1 + 3.93e61T + 1.08e124T^{2} \) |
| 73 | \( 1 - 2.89e62T + 6.96e124T^{2} \) |
| 79 | \( 1 - 3.53e63T + 1.38e127T^{2} \) |
| 83 | \( 1 - 1.40e64T + 3.78e128T^{2} \) |
| 89 | \( 1 - 3.15e65T + 4.06e130T^{2} \) |
| 97 | \( 1 - 7.03e66T + 1.29e133T^{2} \) |
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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
−13.23998512172594077933737401146, −12.00465514542109437259587572920, −10.05622706752211428626373644034, −9.526634750839215239361956002843, −7.79143854470399572267012580349, −6.18171860039340669944070963883, −5.06336051497060170207912828714, −3.69685319425590307447665814527, −1.85811698967749958776715425937, −0.45732055740112826516995209584,
0.45732055740112826516995209584, 1.85811698967749958776715425937, 3.69685319425590307447665814527, 5.06336051497060170207912828714, 6.18171860039340669944070963883, 7.79143854470399572267012580349, 9.526634750839215239361956002843, 10.05622706752211428626373644034, 12.00465514542109437259587572920, 13.23998512172594077933737401146
