L(s) = 1 | + 0.301·2-s − 1.90·4-s + 0.698·5-s + 7-s − 1.17·8-s + 0.210·10-s − 4.43·13-s + 0.301·14-s + 3.46·16-s − 6.80·17-s − 1.78·19-s − 1.33·20-s + 9.20·23-s − 4.51·25-s − 1.33·26-s − 1.90·28-s + 7.89·29-s + 10.4·31-s + 3.39·32-s − 2.04·34-s + 0.698·35-s + 6.16·37-s − 0.538·38-s − 0.822·40-s + 1.33·41-s − 6.33·43-s + 2.77·46-s + ⋯ |
L(s) = 1 | + 0.212·2-s − 0.954·4-s + 0.312·5-s + 0.377·7-s − 0.416·8-s + 0.0665·10-s − 1.22·13-s + 0.0804·14-s + 0.866·16-s − 1.65·17-s − 0.410·19-s − 0.298·20-s + 1.91·23-s − 0.902·25-s − 0.261·26-s − 0.360·28-s + 1.46·29-s + 1.87·31-s + 0.600·32-s − 0.351·34-s + 0.118·35-s + 1.01·37-s − 0.0874·38-s − 0.130·40-s + 0.208·41-s − 0.965·43-s + 0.408·46-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.301T + 2T^{2} \) |
| 5 | \( 1 - 0.698T + 5T^{2} \) |
| 13 | \( 1 + 4.43T + 13T^{2} \) |
| 17 | \( 1 + 6.80T + 17T^{2} \) |
| 19 | \( 1 + 1.78T + 19T^{2} \) |
| 23 | \( 1 - 9.20T + 23T^{2} \) |
| 29 | \( 1 - 7.89T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 6.16T + 37T^{2} \) |
| 41 | \( 1 - 1.33T + 41T^{2} \) |
| 43 | \( 1 + 6.33T + 43T^{2} \) |
| 47 | \( 1 + 8.20T + 47T^{2} \) |
| 53 | \( 1 - 2.35T + 53T^{2} \) |
| 59 | \( 1 - 4.76T + 59T^{2} \) |
| 61 | \( 1 - 1.69T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 + 7.58T + 71T^{2} \) |
| 73 | \( 1 + 6.96T + 73T^{2} \) |
| 79 | \( 1 + 8.85T + 79T^{2} \) |
| 83 | \( 1 - 5.71T + 83T^{2} \) |
| 89 | \( 1 - 2.22T + 89T^{2} \) |
| 97 | \( 1 + 0.171T + 97T^{2} \) |
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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
−7.57089658697639481994708473944, −6.72048755238875825824261180701, −6.17883981924136073356321494190, −5.14118472422754750640558160063, −4.66008900148433388667818920308, −4.29917916464032888887278041384, −3.00399513216261021161541558313, −2.43282157038605634055200820796, −1.17288776228353016214387945567, 0,
1.17288776228353016214387945567, 2.43282157038605634055200820796, 3.00399513216261021161541558313, 4.29917916464032888887278041384, 4.66008900148433388667818920308, 5.14118472422754750640558160063, 6.17883981924136073356321494190, 6.72048755238875825824261180701, 7.57089658697639481994708473944