| L(s) = 1 | − 4.68e11·2-s − 9.12e18·3-s − 3.85e23·4-s + 5.54e27·5-s + 4.27e30·6-s + 1.91e33·7-s + 4.63e35·8-s + 3.39e37·9-s − 2.59e39·10-s − 1.76e41·11-s + 3.51e42·12-s − 1.46e44·13-s − 8.96e44·14-s − 5.05e46·15-s + 1.57e46·16-s + 5.82e48·17-s − 1.58e49·18-s − 1.24e50·19-s − 2.13e51·20-s − 1.74e52·21-s + 8.27e52·22-s − 2.97e53·23-s − 4.22e54·24-s + 1.42e55·25-s + 6.86e55·26-s + 1.39e56·27-s − 7.37e56·28-s + ⋯ |
| L(s) = 1 | − 0.602·2-s − 1.29·3-s − 0.637·4-s + 1.36·5-s + 0.782·6-s + 0.795·7-s + 0.986·8-s + 0.688·9-s − 0.821·10-s − 1.29·11-s + 0.828·12-s − 1.46·13-s − 0.479·14-s − 1.77·15-s + 0.0430·16-s + 1.45·17-s − 0.415·18-s − 0.382·19-s − 0.868·20-s − 1.03·21-s + 0.779·22-s − 0.484·23-s − 1.28·24-s + 0.859·25-s + 0.881·26-s + 0.404·27-s − 0.506·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(80-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+79/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(40)\) |
\(\approx\) |
\(0.7339617647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7339617647\) |
| \(L(\frac{81}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 4.68e11T + 6.04e23T^{2} \) |
| 3 | \( 1 + 9.12e18T + 4.92e37T^{2} \) |
| 5 | \( 1 - 5.54e27T + 1.65e55T^{2} \) |
| 7 | \( 1 - 1.91e33T + 5.79e66T^{2} \) |
| 11 | \( 1 + 1.76e41T + 1.86e82T^{2} \) |
| 13 | \( 1 + 1.46e44T + 1.00e88T^{2} \) |
| 17 | \( 1 - 5.82e48T + 1.60e97T^{2} \) |
| 19 | \( 1 + 1.24e50T + 1.05e101T^{2} \) |
| 23 | \( 1 + 2.97e53T + 3.77e107T^{2} \) |
| 29 | \( 1 + 6.86e57T + 3.38e115T^{2} \) |
| 31 | \( 1 + 2.22e57T + 6.57e117T^{2} \) |
| 37 | \( 1 + 2.44e61T + 7.72e123T^{2} \) |
| 41 | \( 1 + 3.29e63T + 2.56e127T^{2} \) |
| 43 | \( 1 - 5.98e64T + 1.10e129T^{2} \) |
| 47 | \( 1 - 9.09e65T + 1.24e132T^{2} \) |
| 53 | \( 1 + 9.08e67T + 1.65e136T^{2} \) |
| 59 | \( 1 - 1.03e70T + 7.89e139T^{2} \) |
| 61 | \( 1 + 4.36e70T + 1.09e141T^{2} \) |
| 67 | \( 1 - 1.33e72T + 1.81e144T^{2} \) |
| 71 | \( 1 - 7.91e72T + 1.77e146T^{2} \) |
| 73 | \( 1 - 6.71e73T + 1.59e147T^{2} \) |
| 79 | \( 1 - 1.03e75T + 8.17e149T^{2} \) |
| 83 | \( 1 + 2.29e75T + 4.04e151T^{2} \) |
| 89 | \( 1 - 1.62e76T + 1.00e154T^{2} \) |
| 97 | \( 1 - 4.52e78T + 9.01e156T^{2} \) |
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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
−16.86619653186549595217117997145, −14.20925602883526194668420297141, −12.58220160359919811499612335053, −10.63408488941928430211627238207, −9.714555013799762722427996299638, −7.71644014095573310098993320260, −5.58665963502690916884971894517, −4.98732340477794019916006972057, −2.02321218593275360637280896813, −0.59352203405527027543619141746,
0.59352203405527027543619141746, 2.02321218593275360637280896813, 4.98732340477794019916006972057, 5.58665963502690916884971894517, 7.71644014095573310098993320260, 9.714555013799762722427996299638, 10.63408488941928430211627238207, 12.58220160359919811499612335053, 14.20925602883526194668420297141, 16.86619653186549595217117997145
