| L(s) = 1 | − 7.75·2-s − 67.8·4-s − 125·5-s − 343·7-s + 1.51e3·8-s + 969.·10-s + 1.43e3·11-s + 1.23e4·13-s + 2.65e3·14-s − 3.09e3·16-s − 2.65e3·17-s + 1.30e3·19-s + 8.48e3·20-s − 1.11e4·22-s − 1.58e4·23-s + 1.56e4·25-s − 9.54e4·26-s + 2.32e4·28-s − 6.79e4·29-s − 5.41e4·31-s − 1.70e5·32-s + 2.05e4·34-s + 4.28e4·35-s − 4.49e5·37-s − 1.01e4·38-s − 1.89e5·40-s + 4.46e5·41-s + ⋯ |
| L(s) = 1 | − 0.685·2-s − 0.530·4-s − 0.447·5-s − 0.377·7-s + 1.04·8-s + 0.306·10-s + 0.324·11-s + 1.55·13-s + 0.259·14-s − 0.188·16-s − 0.130·17-s + 0.0437·19-s + 0.237·20-s − 0.222·22-s − 0.270·23-s + 0.199·25-s − 1.06·26-s + 0.200·28-s − 0.517·29-s − 0.326·31-s − 0.919·32-s + 0.0897·34-s + 0.169·35-s − 1.45·37-s − 0.0300·38-s − 0.469·40-s + 1.01·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.9413279150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9413279150\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 125T \) |
| 7 | \( 1 + 343T \) |
| good | 2 | \( 1 + 7.75T + 128T^{2} \) |
| 11 | \( 1 - 1.43e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.23e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.65e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.30e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.58e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 6.79e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 5.41e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.49e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.46e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.52e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.20e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.27e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 6.41e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 8.58e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.54e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.67e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 8.20e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.62e4T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.01e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.14e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.34e6T + 8.07e13T^{2} \) |
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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
−10.41370565970553068675526942759, −9.306347748415991886737401483958, −8.688917865040797500647866732701, −7.83191179734260908745327707512, −6.74993921542600298401372476392, −5.57460409279798218200141075805, −4.23419852979626519569508299997, −3.44769638363214158363206497709, −1.62647513179100728456506704077, −0.55085386584939366138429180271,
0.55085386584939366138429180271, 1.62647513179100728456506704077, 3.44769638363214158363206497709, 4.23419852979626519569508299997, 5.57460409279798218200141075805, 6.74993921542600298401372476392, 7.83191179734260908745327707512, 8.688917865040797500647866732701, 9.306347748415991886737401483958, 10.41370565970553068675526942759
