| L(s) = 1 | − 2.43·2-s + 3.90·4-s − 1.43·5-s − 7-s − 4.64·8-s + 3.47·10-s + 2.30·13-s + 2.43·14-s + 3.46·16-s − 1.14·17-s + 5.47·19-s − 5.59·20-s − 3.00·23-s − 2.95·25-s − 5.59·26-s − 3.90·28-s + 3.16·29-s − 6.99·31-s + 0.861·32-s + 2.78·34-s + 1.43·35-s − 8.16·37-s − 13.3·38-s + 6.64·40-s − 5.59·41-s + 10.5·43-s + 7.30·46-s + ⋯ |
| L(s) = 1 | − 1.71·2-s + 1.95·4-s − 0.639·5-s − 0.377·7-s − 1.64·8-s + 1.09·10-s + 0.638·13-s + 0.649·14-s + 0.866·16-s − 0.277·17-s + 1.25·19-s − 1.25·20-s − 0.626·23-s − 0.590·25-s − 1.09·26-s − 0.738·28-s + 0.588·29-s − 1.25·31-s + 0.152·32-s + 0.476·34-s + 0.241·35-s − 1.34·37-s − 2.16·38-s + 1.05·40-s − 0.873·41-s + 1.61·43-s + 1.07·46-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
| \(L(1)\) |
\(\approx\) |
\(0.5217684010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5217684010\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.43T + 2T^{2} \) |
| 5 | \( 1 + 1.43T + 5T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 + 1.14T + 17T^{2} \) |
| 19 | \( 1 - 5.47T + 19T^{2} \) |
| 23 | \( 1 + 3.00T + 23T^{2} \) |
| 29 | \( 1 - 3.16T + 29T^{2} \) |
| 31 | \( 1 + 6.99T + 31T^{2} \) |
| 37 | \( 1 + 8.16T + 37T^{2} \) |
| 41 | \( 1 + 5.59T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 + 9.28T + 53T^{2} \) |
| 59 | \( 1 - 6.89T + 59T^{2} \) |
| 61 | \( 1 + 8.50T + 61T^{2} \) |
| 67 | \( 1 - 7.61T + 67T^{2} \) |
| 71 | \( 1 - 1.92T + 71T^{2} \) |
| 73 | \( 1 - 4.83T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 9.40T + 83T^{2} \) |
| 89 | \( 1 - 4.35T + 89T^{2} \) |
| 97 | \( 1 + 6.56T + 97T^{2} \) |
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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
−7.917382364682249213458417672477, −7.42858880310223811514609863497, −6.84251800854045768419079400227, −6.07076663609294732899623002093, −5.28131912055786420394527595784, −4.05476586550508274327281657198, −3.39249599567391088803258740906, −2.39634956127631973810816390912, −1.46549675322359722394018249135, −0.48999723273947608286955237481,
0.48999723273947608286955237481, 1.46549675322359722394018249135, 2.39634956127631973810816390912, 3.39249599567391088803258740906, 4.05476586550508274327281657198, 5.28131912055786420394527595784, 6.07076663609294732899623002093, 6.84251800854045768419079400227, 7.42858880310223811514609863497, 7.917382364682249213458417672477
