L(s) = 1 | + 2-s + 4-s + 1.41·5-s − 1.27·7-s + 8-s + 1.41·10-s + 5.29·11-s + 6.20·13-s − 1.27·14-s + 16-s + 6.19·17-s + 4.29·19-s + 1.41·20-s + 5.29·22-s − 6.65·23-s − 2.99·25-s + 6.20·26-s − 1.27·28-s + 4.65·29-s + 0.972·31-s + 32-s + 6.19·34-s − 1.80·35-s + 0.986·37-s + 4.29·38-s + 1.41·40-s − 8.34·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.633·5-s − 0.482·7-s + 0.353·8-s + 0.448·10-s + 1.59·11-s + 1.72·13-s − 0.340·14-s + 0.250·16-s + 1.50·17-s + 0.984·19-s + 0.316·20-s + 1.12·22-s − 1.38·23-s − 0.598·25-s + 1.21·26-s − 0.241·28-s + 0.864·29-s + 0.174·31-s + 0.176·32-s + 1.06·34-s − 0.305·35-s + 0.162·37-s + 0.696·38-s + 0.224·40-s − 1.30·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.861292777\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.861292777\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 + 1.27T + 7T^{2} \) |
| 11 | \( 1 - 5.29T + 11T^{2} \) |
| 13 | \( 1 - 6.20T + 13T^{2} \) |
| 17 | \( 1 - 6.19T + 17T^{2} \) |
| 19 | \( 1 - 4.29T + 19T^{2} \) |
| 23 | \( 1 + 6.65T + 23T^{2} \) |
| 29 | \( 1 - 4.65T + 29T^{2} \) |
| 31 | \( 1 - 0.972T + 31T^{2} \) |
| 37 | \( 1 - 0.986T + 37T^{2} \) |
| 41 | \( 1 + 8.34T + 41T^{2} \) |
| 43 | \( 1 - 3.52T + 43T^{2} \) |
| 47 | \( 1 - 4.00T + 47T^{2} \) |
| 53 | \( 1 + 0.663T + 53T^{2} \) |
| 59 | \( 1 + 4.27T + 59T^{2} \) |
| 61 | \( 1 - 3.90T + 61T^{2} \) |
| 67 | \( 1 - 2.70T + 67T^{2} \) |
| 71 | \( 1 + 2.68T + 71T^{2} \) |
| 73 | \( 1 + 4.49T + 73T^{2} \) |
| 79 | \( 1 + 5.09T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 + 8.01T + 89T^{2} \) |
| 97 | \( 1 + 2.43T + 97T^{2} \) |
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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
−7.79309581188082295433574067562, −6.86108773549763061507203314982, −6.16083530421418376370202412745, −5.98070020694508516548248906795, −5.16693319516096008560117272039, −3.99170190436260107619219725599, −3.68803986224519838144361367342, −2.91844293653560055605162191922, −1.63960047149821116377783565681, −1.12919971501791974702763863626,
1.12919971501791974702763863626, 1.63960047149821116377783565681, 2.91844293653560055605162191922, 3.68803986224519838144361367342, 3.99170190436260107619219725599, 5.16693319516096008560117272039, 5.98070020694508516548248906795, 6.16083530421418376370202412745, 6.86108773549763061507203314982, 7.79309581188082295433574067562