| L(s) = 1 | + 2.76·2-s + 5.62·4-s + 2.62·5-s − 7-s + 10.0·8-s + 7.25·10-s + 2.38·13-s − 2.76·14-s + 16.4·16-s − 2.38·17-s + 1.72·19-s + 14.7·20-s + 0.626·23-s + 1.89·25-s + 6.59·26-s − 5.62·28-s + 1.72·29-s − 2.23·31-s + 25.2·32-s − 6.59·34-s − 2.62·35-s − 6.89·37-s + 4.77·38-s + 26.2·40-s + 10.3·41-s − 7.25·43-s + 1.72·46-s + ⋯ |
| L(s) = 1 | + 1.95·2-s + 2.81·4-s + 1.17·5-s − 0.377·7-s + 3.54·8-s + 2.29·10-s + 0.662·13-s − 0.738·14-s + 4.10·16-s − 0.579·17-s + 0.396·19-s + 3.30·20-s + 0.130·23-s + 0.379·25-s + 1.29·26-s − 1.06·28-s + 0.321·29-s − 0.402·31-s + 4.46·32-s − 1.13·34-s − 0.443·35-s − 1.13·37-s + 0.774·38-s + 4.15·40-s + 1.62·41-s − 1.10·43-s + 0.254·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.27241886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.27241886\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.76T + 2T^{2} \) |
| 5 | \( 1 - 2.62T + 5T^{2} \) |
| 13 | \( 1 - 2.38T + 13T^{2} \) |
| 17 | \( 1 + 2.38T + 17T^{2} \) |
| 19 | \( 1 - 1.72T + 19T^{2} \) |
| 23 | \( 1 - 0.626T + 23T^{2} \) |
| 29 | \( 1 - 1.72T + 29T^{2} \) |
| 31 | \( 1 + 2.23T + 31T^{2} \) |
| 37 | \( 1 + 6.89T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 7.25T + 43T^{2} \) |
| 47 | \( 1 + 6.38T + 47T^{2} \) |
| 53 | \( 1 - 9.25T + 53T^{2} \) |
| 59 | \( 1 - 1.76T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 6.42T + 67T^{2} \) |
| 71 | \( 1 + 8.08T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + 8.35T + 97T^{2} \) |
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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
−7.41605524640052492976238567371, −6.80842841993253907657788966388, −6.23498916662096590302486849302, −5.69344980700897458412140715690, −5.17364036238851812261610415893, −4.34447642194382639237273904174, −3.60938147984727669667911008594, −2.86720112894506222377436063802, −2.13387444342417116804663212635, −1.35962232738039297376221383276,
1.35962232738039297376221383276, 2.13387444342417116804663212635, 2.86720112894506222377436063802, 3.60938147984727669667911008594, 4.34447642194382639237273904174, 5.17364036238851812261610415893, 5.69344980700897458412140715690, 6.23498916662096590302486849302, 6.80842841993253907657788966388, 7.41605524640052492976238567371
