L(s) = 1 | + 2.48·2-s + 4.19·4-s + 5-s − 1.19·7-s + 5.46·8-s + 2.48·10-s + 1.19·11-s + 13-s − 2.97·14-s + 5.21·16-s − 6.17·17-s + 6.97·19-s + 4.19·20-s + 2.97·22-s − 4.17·23-s + 25-s + 2.48·26-s − 5.02·28-s − 6·29-s − 2.97·31-s + 2.05·32-s − 15.3·34-s − 1.19·35-s + 7.78·37-s + 17.3·38-s + 5.46·40-s + 6.17·41-s + ⋯ |
L(s) = 1 | + 1.76·2-s + 2.09·4-s + 0.447·5-s − 0.452·7-s + 1.93·8-s + 0.787·10-s + 0.360·11-s + 0.277·13-s − 0.796·14-s + 1.30·16-s − 1.49·17-s + 1.60·19-s + 0.938·20-s + 0.635·22-s − 0.870·23-s + 0.200·25-s + 0.488·26-s − 0.948·28-s − 1.11·29-s − 0.534·31-s + 0.362·32-s − 2.63·34-s − 0.202·35-s + 1.27·37-s + 2.81·38-s + 0.864·40-s + 0.964·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.042476810\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.042476810\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 - 2.48T + 2T^{2} \) |
| 7 | \( 1 + 1.19T + 7T^{2} \) |
| 11 | \( 1 - 1.19T + 11T^{2} \) |
| 17 | \( 1 + 6.17T + 17T^{2} \) |
| 19 | \( 1 - 6.97T + 19T^{2} \) |
| 23 | \( 1 + 4.17T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 2.97T + 31T^{2} \) |
| 37 | \( 1 - 7.78T + 37T^{2} \) |
| 41 | \( 1 - 6.17T + 41T^{2} \) |
| 43 | \( 1 + 9.95T + 43T^{2} \) |
| 47 | \( 1 - 1.02T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 5.37T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 9.37T + 67T^{2} \) |
| 71 | \( 1 - 5.19T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 + 1.78T + 79T^{2} \) |
| 83 | \( 1 - 5.37T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 + 1.82T + 97T^{2} \) |
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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
−11.28947496160790155345587643501, −9.949136806755564029889054996514, −9.106966165072204819750037331354, −7.64317386362695216035855708318, −6.64833433437964352001168506392, −6.01428354362857075115516384799, −5.10361484874060257198783181453, −4.08345915782330000823802932237, −3.15847293670893930892454280812, −1.96140697641443756608236313913,
1.96140697641443756608236313913, 3.15847293670893930892454280812, 4.08345915782330000823802932237, 5.10361484874060257198783181453, 6.01428354362857075115516384799, 6.64833433437964352001168506392, 7.64317386362695216035855708318, 9.106966165072204819750037331354, 9.949136806755564029889054996514, 11.28947496160790155345587643501