| L(s) = 1 | + 6.87e10·2-s + 2.54e17·3-s + 4.72e21·4-s + 6.33e25·5-s + 1.74e28·6-s + 7.98e30·7-s + 3.24e32·8-s − 3.00e33·9-s + 4.35e36·10-s + 7.13e37·11-s + 1.20e39·12-s + 4.11e40·13-s + 5.48e41·14-s + 1.60e43·15-s + 2.23e43·16-s − 4.48e43·17-s − 2.06e44·18-s − 2.90e46·19-s + 2.99e47·20-s + 2.03e48·21-s + 4.90e48·22-s − 7.33e49·23-s + 8.24e49·24-s + 2.95e51·25-s + 2.82e51·26-s − 1.79e52·27-s + 3.77e52·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.977·3-s + 0.5·4-s + 1.94·5-s + 0.691·6-s + 1.13·7-s + 0.353·8-s − 0.0444·9-s + 1.37·10-s + 0.696·11-s + 0.488·12-s + 0.902·13-s + 0.805·14-s + 1.90·15-s + 0.250·16-s − 0.0549·17-s − 0.0314·18-s − 0.615·19-s + 0.973·20-s + 1.11·21-s + 0.492·22-s − 1.45·23-s + 0.345·24-s + 2.78·25-s + 0.638·26-s − 1.02·27-s + 0.569·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(74-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+73/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(37)\) |
\(\approx\) |
\(7.880932683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.880932683\) |
| \(L(\frac{75}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 6.87e10T \) |
| good | 3 | \( 1 - 2.54e17T + 6.75e34T^{2} \) |
| 5 | \( 1 - 6.33e25T + 1.05e51T^{2} \) |
| 7 | \( 1 - 7.98e30T + 4.92e61T^{2} \) |
| 11 | \( 1 - 7.13e37T + 1.05e76T^{2} \) |
| 13 | \( 1 - 4.11e40T + 2.07e81T^{2} \) |
| 17 | \( 1 + 4.48e43T + 6.64e89T^{2} \) |
| 19 | \( 1 + 2.90e46T + 2.23e93T^{2} \) |
| 23 | \( 1 + 7.33e49T + 2.54e99T^{2} \) |
| 29 | \( 1 + 3.82e53T + 5.68e106T^{2} \) |
| 31 | \( 1 + 2.83e54T + 7.40e108T^{2} \) |
| 37 | \( 1 - 1.25e56T + 3.01e114T^{2} \) |
| 41 | \( 1 + 6.14e58T + 5.41e117T^{2} \) |
| 43 | \( 1 + 2.13e59T + 1.75e119T^{2} \) |
| 47 | \( 1 + 1.57e61T + 1.15e122T^{2} \) |
| 53 | \( 1 + 9.46e62T + 7.44e125T^{2} \) |
| 59 | \( 1 - 4.99e63T + 1.87e129T^{2} \) |
| 61 | \( 1 - 1.28e65T + 2.13e130T^{2} \) |
| 67 | \( 1 - 4.85e66T + 2.01e133T^{2} \) |
| 71 | \( 1 - 9.31e66T + 1.38e135T^{2} \) |
| 73 | \( 1 - 1.98e68T + 1.05e136T^{2} \) |
| 79 | \( 1 - 2.37e68T + 3.36e138T^{2} \) |
| 83 | \( 1 - 1.57e70T + 1.23e140T^{2} \) |
| 89 | \( 1 + 2.10e71T + 2.02e142T^{2} \) |
| 97 | \( 1 - 3.81e72T + 1.08e145T^{2} \) |
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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
−14.10675563769391858642041409662, −13.14946684537489262745047684154, −11.11986015750992233779885953984, −9.526751879706395336219358470331, −8.306470530738937799334742129410, −6.35118098446255115275391162346, −5.28773759088744606405104235771, −3.63664937890014308213176451388, −2.02070197058076735062048504161, −1.71203247275573186349192304432,
1.71203247275573186349192304432, 2.02070197058076735062048504161, 3.63664937890014308213176451388, 5.28773759088744606405104235771, 6.35118098446255115275391162346, 8.306470530738937799334742129410, 9.526751879706395336219358470331, 11.11986015750992233779885953984, 13.14946684537489262745047684154, 14.10675563769391858642041409662
