| L(s) = 1 | − 2.45·2-s + 4.04·4-s − 2.05·5-s + 7-s − 5.01·8-s + 5.04·10-s + 0.206·13-s − 2.45·14-s + 4.24·16-s − 0.813·17-s + 6.28·19-s − 8.28·20-s + 5.11·23-s − 0.793·25-s − 0.507·26-s + 4.04·28-s + 3.87·29-s + 6.49·31-s − 0.406·32-s + 2·34-s − 2.05·35-s + 1.79·37-s − 15.4·38-s + 10.2·40-s + 1.82·41-s + 10.0·43-s − 12.5·46-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 2.02·4-s − 0.917·5-s + 0.377·7-s − 1.77·8-s + 1.59·10-s + 0.0572·13-s − 0.656·14-s + 1.06·16-s − 0.197·17-s + 1.44·19-s − 1.85·20-s + 1.06·23-s − 0.158·25-s − 0.0995·26-s + 0.763·28-s + 0.720·29-s + 1.16·31-s − 0.0719·32-s + 0.342·34-s − 0.346·35-s + 0.294·37-s − 2.50·38-s + 1.62·40-s + 0.285·41-s + 1.53·43-s − 1.85·46-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
| \(L(1)\) |
\(\approx\) |
\(0.7787526018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7787526018\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.45T + 2T^{2} \) |
| 5 | \( 1 + 2.05T + 5T^{2} \) |
| 13 | \( 1 - 0.206T + 13T^{2} \) |
| 17 | \( 1 + 0.813T + 17T^{2} \) |
| 19 | \( 1 - 6.28T + 19T^{2} \) |
| 23 | \( 1 - 5.11T + 23T^{2} \) |
| 29 | \( 1 - 3.87T + 29T^{2} \) |
| 31 | \( 1 - 6.49T + 31T^{2} \) |
| 37 | \( 1 - 1.79T + 37T^{2} \) |
| 41 | \( 1 - 1.82T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 + 8.79T + 47T^{2} \) |
| 53 | \( 1 + 3.08T + 53T^{2} \) |
| 59 | \( 1 - 2.66T + 59T^{2} \) |
| 61 | \( 1 - 3.58T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 - 7.79T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + 5.72T + 89T^{2} \) |
| 97 | \( 1 + 8.49T + 97T^{2} \) |
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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
−7.988594349330098611170474895778, −7.43910427694957814545631010586, −6.90871976959521842386226153868, −6.10730232309077684242299965610, −5.08891390278812808656898960444, −4.28327835771297406229783939471, −3.21784671379191344228315577512, −2.50741391368027020748183598654, −1.32092205019387171112824536178, −0.64900915382474614952383272758,
0.64900915382474614952383272758, 1.32092205019387171112824536178, 2.50741391368027020748183598654, 3.21784671379191344228315577512, 4.28327835771297406229783939471, 5.08891390278812808656898960444, 6.10730232309077684242299965610, 6.90871976959521842386226153868, 7.43910427694957814545631010586, 7.988594349330098611170474895778
