| L(s) = 1 | − 2.19e12·2-s − 7.32e17·3-s + 4.83e24·4-s − 1.23e29·5-s + 1.61e30·6-s + 6.61e34·7-s − 1.06e37·8-s − 3.99e39·9-s + 2.72e41·10-s − 2.35e43·11-s − 3.54e42·12-s + 2.17e46·13-s − 1.45e47·14-s + 9.06e46·15-s + 2.33e49·16-s − 2.06e51·17-s + 8.77e51·18-s − 1.54e53·19-s − 5.98e53·20-s − 4.84e52·21-s + 5.17e55·22-s + 3.09e56·23-s + 7.78e54·24-s + 4.96e57·25-s − 4.78e58·26-s + 5.84e57·27-s + 3.19e59·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0115·3-s + 0.5·4-s − 1.21·5-s + 0.00819·6-s + 0.561·7-s − 0.353·8-s − 0.999·9-s + 0.860·10-s − 1.42·11-s − 0.00579·12-s + 1.28·13-s − 0.396·14-s + 0.0141·15-s + 0.250·16-s − 1.77·17-s + 0.707·18-s − 1.31·19-s − 0.608·20-s − 0.00650·21-s + 1.00·22-s + 0.953·23-s + 0.00409·24-s + 0.480·25-s − 0.908·26-s + 0.0231·27-s + 0.280·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(84-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+83/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(42)\) |
\(\approx\) |
\(0.2684754695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2684754695\) |
| \(L(\frac{85}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.19e12T \) |
| good | 3 | \( 1 + 7.32e17T + 3.99e39T^{2} \) |
| 5 | \( 1 + 1.23e29T + 1.03e58T^{2} \) |
| 7 | \( 1 - 6.61e34T + 1.39e70T^{2} \) |
| 11 | \( 1 + 2.35e43T + 2.72e86T^{2} \) |
| 13 | \( 1 - 2.17e46T + 2.86e92T^{2} \) |
| 17 | \( 1 + 2.06e51T + 1.34e102T^{2} \) |
| 19 | \( 1 + 1.54e53T + 1.36e106T^{2} \) |
| 23 | \( 1 - 3.09e56T + 1.05e113T^{2} \) |
| 29 | \( 1 + 4.67e59T + 2.39e121T^{2} \) |
| 31 | \( 1 + 1.70e61T + 6.06e123T^{2} \) |
| 37 | \( 1 + 9.03e64T + 1.44e130T^{2} \) |
| 41 | \( 1 + 8.21e66T + 7.26e133T^{2} \) |
| 43 | \( 1 + 5.19e67T + 3.78e135T^{2} \) |
| 47 | \( 1 + 3.72e69T + 6.08e138T^{2} \) |
| 53 | \( 1 + 4.87e71T + 1.30e143T^{2} \) |
| 59 | \( 1 - 2.59e73T + 9.56e146T^{2} \) |
| 61 | \( 1 + 2.95e73T + 1.52e148T^{2} \) |
| 67 | \( 1 - 9.82e75T + 3.66e151T^{2} \) |
| 71 | \( 1 + 2.74e76T + 4.51e153T^{2} \) |
| 73 | \( 1 - 1.97e77T + 4.52e154T^{2} \) |
| 79 | \( 1 - 2.44e78T + 3.18e157T^{2} \) |
| 83 | \( 1 + 4.50e79T + 1.92e159T^{2} \) |
| 89 | \( 1 - 1.00e81T + 6.30e161T^{2} \) |
| 97 | \( 1 - 2.37e82T + 7.98e164T^{2} \) |
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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
−13.12488263780520650151726988858, −11.31778640454824613380333939006, −10.88167920773412221064390714451, −8.593948661947357553317514111734, −8.157539316703607536004586605043, −6.58905180427988984388776399401, −4.89231643953424577113814838626, −3.36477498500808209903509353717, −2.01410371192593249767918229774, −0.27143987742286290018096475355,
0.27143987742286290018096475355, 2.01410371192593249767918229774, 3.36477498500808209903509353717, 4.89231643953424577113814838626, 6.58905180427988984388776399401, 8.157539316703607536004586605043, 8.593948661947357553317514111734, 10.88167920773412221064390714451, 11.31778640454824613380333939006, 13.12488263780520650151726988858
