| L(s) = 1 | + 2.06e10·2-s − 5.55e15·3-s + 2.80e20·4-s + 4.26e23·5-s − 1.14e26·6-s − 3.03e28·7-s + 2.74e30·8-s + 3.09e31·9-s + 8.83e33·10-s − 3.14e34·11-s − 1.55e36·12-s + 2.32e37·13-s − 6.27e38·14-s − 2.37e39·15-s + 1.54e40·16-s + 2.99e41·17-s + 6.39e41·18-s − 2.19e42·19-s + 1.19e44·20-s + 1.68e44·21-s − 6.49e44·22-s + 5.39e45·23-s − 1.52e46·24-s + 1.14e47·25-s + 4.80e47·26-s − 1.71e47·27-s − 8.50e48·28-s + ⋯ |
| L(s) = 1 | + 1.70·2-s − 0.577·3-s + 1.89·4-s + 1.64·5-s − 0.983·6-s − 1.48·7-s + 1.53·8-s + 0.333·9-s + 2.79·10-s − 0.407·11-s − 1.09·12-s + 1.11·13-s − 2.52·14-s − 0.947·15-s + 0.709·16-s + 1.80·17-s + 0.567·18-s − 0.318·19-s + 3.11·20-s + 0.855·21-s − 0.694·22-s + 1.30·23-s − 0.884·24-s + 1.69·25-s + 1.90·26-s − 0.192·27-s − 2.81·28-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
| \(L(34)\) |
\(\approx\) |
\(6.762861369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.762861369\) |
| \(L(\frac{69}{2})\) |
|
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 | \( 1 - 2.06e10T + 1.47e20T^{2} \) |
| 5 | \( 1 - 4.26e23T + 6.77e46T^{2} \) |
| 7 | \( 1 + 3.03e28T + 4.18e56T^{2} \) |
| 11 | \( 1 + 3.14e34T + 5.93e69T^{2} \) |
| 13 | \( 1 - 2.32e37T + 4.30e74T^{2} \) |
| 17 | \( 1 - 2.99e41T + 2.75e82T^{2} \) |
| 19 | \( 1 + 2.19e42T + 4.74e85T^{2} \) |
| 23 | \( 1 - 5.39e45T + 1.72e91T^{2} \) |
| 29 | \( 1 - 1.03e48T + 9.56e97T^{2} \) |
| 31 | \( 1 + 1.03e49T + 8.34e99T^{2} \) |
| 37 | \( 1 + 7.45e51T + 1.17e105T^{2} \) |
| 41 | \( 1 - 1.59e54T + 1.13e108T^{2} \) |
| 43 | \( 1 + 3.44e54T + 2.76e109T^{2} \) |
| 47 | \( 1 - 1.39e56T + 1.07e112T^{2} \) |
| 53 | \( 1 + 7.64e57T + 3.36e115T^{2} \) |
| 59 | \( 1 - 3.63e58T + 4.43e118T^{2} \) |
| 61 | \( 1 - 4.97e59T + 4.14e119T^{2} \) |
| 67 | \( 1 - 2.19e61T + 2.22e122T^{2} \) |
| 71 | \( 1 - 1.04e62T + 1.08e124T^{2} \) |
| 73 | \( 1 - 2.78e62T + 6.96e124T^{2} \) |
| 79 | \( 1 + 3.71e63T + 1.38e127T^{2} \) |
| 83 | \( 1 + 3.08e62T + 3.78e128T^{2} \) |
| 89 | \( 1 + 1.40e65T + 4.06e130T^{2} \) |
| 97 | \( 1 + 1.11e66T + 1.29e133T^{2} \) |
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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
−13.15154253122554763755529337601, −12.59090717091238506080812883277, −10.76096165110461843381885299542, −9.573716298405564693596227753895, −6.73945595816369427424580215197, −5.93746334467265145284643082508, −5.31092524522969633570688180997, −3.56675364695678410427858841213, −2.59996056363700573207562719676, −1.10761350216465096631539876515,
1.10761350216465096631539876515, 2.59996056363700573207562719676, 3.56675364695678410427858841213, 5.31092524522969633570688180997, 5.93746334467265145284643082508, 6.73945595816369427424580215197, 9.573716298405564693596227753895, 10.76096165110461843381885299542, 12.59090717091238506080812883277, 13.15154253122554763755529337601
