| L(s) = 1 | − 1.60·2-s + 0.568·4-s − 1.76·5-s − 7-s + 2.29·8-s + 2.82·10-s + 0.237·13-s + 1.60·14-s − 4.81·16-s + 6.14·17-s + 1.53·19-s − 1.00·20-s − 3.89·23-s − 1.88·25-s − 0.381·26-s − 0.568·28-s − 4.45·29-s + 8.52·31-s + 3.12·32-s − 9.84·34-s + 1.76·35-s − 3.64·37-s − 2.46·38-s − 4.04·40-s − 1.90·41-s − 11.7·43-s + 6.24·46-s + ⋯ |
| L(s) = 1 | − 1.13·2-s + 0.284·4-s − 0.789·5-s − 0.377·7-s + 0.811·8-s + 0.894·10-s + 0.0659·13-s + 0.428·14-s − 1.20·16-s + 1.48·17-s + 0.352·19-s − 0.224·20-s − 0.812·23-s − 0.377·25-s − 0.0747·26-s − 0.107·28-s − 0.827·29-s + 1.53·31-s + 0.552·32-s − 1.68·34-s + 0.298·35-s − 0.599·37-s − 0.399·38-s − 0.640·40-s − 0.297·41-s − 1.78·43-s + 0.920·46-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 | \( 1 + 1.60T + 2T^{2} \) |
| 5 | \( 1 + 1.76T + 5T^{2} \) |
| 13 | \( 1 - 0.237T + 13T^{2} \) |
| 17 | \( 1 - 6.14T + 17T^{2} \) |
| 19 | \( 1 - 1.53T + 19T^{2} \) |
| 23 | \( 1 + 3.89T + 23T^{2} \) |
| 29 | \( 1 + 4.45T + 29T^{2} \) |
| 31 | \( 1 - 8.52T + 31T^{2} \) |
| 37 | \( 1 + 3.64T + 37T^{2} \) |
| 41 | \( 1 + 1.90T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 8.55T + 47T^{2} \) |
| 53 | \( 1 + 3.14T + 53T^{2} \) |
| 59 | \( 1 - 2.56T + 59T^{2} \) |
| 61 | \( 1 + 6.78T + 61T^{2} \) |
| 67 | \( 1 + 1.62T + 67T^{2} \) |
| 71 | \( 1 - 7.37T + 71T^{2} \) |
| 73 | \( 1 - 4.60T + 73T^{2} \) |
| 79 | \( 1 + 17.2T + 79T^{2} \) |
| 83 | \( 1 - 9.16T + 83T^{2} \) |
| 89 | \( 1 - 1.14T + 89T^{2} \) |
| 97 | \( 1 + 4.68T + 97T^{2} \) |
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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
−7.71345323941883064827335986472, −7.19255136147686045910148354198, −6.31601291653004253225558710299, −5.47693323004495475872141450079, −4.63880557609886518601786689580, −3.80762989853085136835631891393, −3.18398942524112759144165677929, −1.94844355894843665033444127406, −0.974535851661545281747643310737, 0,
0.974535851661545281747643310737, 1.94844355894843665033444127406, 3.18398942524112759144165677929, 3.80762989853085136835631891393, 4.63880557609886518601786689580, 5.47693323004495475872141450079, 6.31601291653004253225558710299, 7.19255136147686045910148354198, 7.71345323941883064827335986472
