| L(s) = 1 | + 19.9·2-s − 76.4·3-s + 270.·4-s + 522.·5-s − 1.52e3·6-s + 653.·7-s + 2.83e3·8-s + 3.65e3·9-s + 1.04e4·10-s − 4.33e3·11-s − 2.06e4·12-s − 1.03e3·13-s + 1.30e4·14-s − 3.99e4·15-s + 2.20e4·16-s − 4.91e3·17-s + 7.29e4·18-s − 1.65e4·19-s + 1.41e5·20-s − 4.99e4·21-s − 8.65e4·22-s − 1.21e4·23-s − 2.16e5·24-s + 1.94e5·25-s − 2.07e4·26-s − 1.12e5·27-s + 1.76e5·28-s + ⋯ |
| L(s) = 1 | + 1.76·2-s − 1.63·3-s + 2.11·4-s + 1.86·5-s − 2.88·6-s + 0.720·7-s + 1.95·8-s + 1.67·9-s + 3.29·10-s − 0.982·11-s − 3.45·12-s − 0.131·13-s + 1.27·14-s − 3.05·15-s + 1.34·16-s − 0.242·17-s + 2.94·18-s − 0.554·19-s + 3.94·20-s − 1.17·21-s − 1.73·22-s − 0.208·23-s − 3.20·24-s + 2.49·25-s − 0.231·26-s − 1.09·27-s + 1.52·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.480837952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.480837952\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 - 19.9T + 128T^{2} \) |
| 3 | \( 1 + 76.4T + 2.18e3T^{2} \) |
| 5 | \( 1 - 522.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 653.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.33e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.03e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 4.91e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.65e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 1.05e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 6.79e3T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.07e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.14e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.07e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.07e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 8.81e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.60e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.21e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 7.83e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.54e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.41e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.43e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.13e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.11e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.08e6T + 8.07e13T^{2} \) |
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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
−16.21598098620972702321722974187, −14.69717021553809380250040745683, −13.43395145994702219473435094054, −12.65719367177306433950605715443, −11.26032948431277262928526960215, −10.30111110503516797665916217096, −6.64304787831468955170627867474, −5.58388356391192602687096004770, −4.95411654581519345186945105889, −1.99628855199381729590217694891,
1.99628855199381729590217694891, 4.95411654581519345186945105889, 5.58388356391192602687096004770, 6.64304787831468955170627867474, 10.30111110503516797665916217096, 11.26032948431277262928526960215, 12.65719367177306433950605715443, 13.43395145994702219473435094054, 14.69717021553809380250040745683, 16.21598098620972702321722974187
