| L(s) = 1 | + 26.7·2-s + 66.4·3-s + 204.·4-s + 625·5-s + 1.77e3·6-s − 5.94e3·7-s − 8.23e3·8-s − 1.52e4·9-s + 1.67e4·10-s + 7.23e4·11-s + 1.35e4·12-s + 1.45e4·13-s − 1.59e5·14-s + 4.15e4·15-s − 3.25e5·16-s + 6.14e5·17-s − 4.08e5·18-s − 2.38e5·19-s + 1.27e5·20-s − 3.95e5·21-s + 1.93e6·22-s + 9.15e4·23-s − 5.47e5·24-s + 3.90e5·25-s + 3.89e5·26-s − 2.32e6·27-s − 1.21e6·28-s + ⋯ |
| L(s) = 1 | + 1.18·2-s + 0.473·3-s + 0.399·4-s + 0.447·5-s + 0.560·6-s − 0.936·7-s − 0.710·8-s − 0.775·9-s + 0.528·10-s + 1.48·11-s + 0.189·12-s + 0.141·13-s − 1.10·14-s + 0.211·15-s − 1.23·16-s + 1.78·17-s − 0.917·18-s − 0.419·19-s + 0.178·20-s − 0.443·21-s + 1.76·22-s + 0.0682·23-s − 0.336·24-s + 0.200·25-s + 0.167·26-s − 0.841·27-s − 0.373·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.307897582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.307897582\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 625T \) |
| good | 2 | \( 1 - 26.7T + 512T^{2} \) |
| 3 | \( 1 - 66.4T + 1.96e4T^{2} \) |
| 7 | \( 1 + 5.94e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.23e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.45e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 6.14e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.38e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 9.15e4T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.23e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.72e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.03e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.84e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.72e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.25e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.66e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.65e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.59e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 7.66e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.03e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.20e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.99e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.13e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.00e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 4.02e8T + 7.60e17T^{2} \) |
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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
−22.21199405941046639495055273162, −20.75174794706359464723370264148, −19.17096478842570809276102939166, −16.91637363899346243655810978992, −14.77087689827306271913144446404, −13.74343807720023324069913020571, −12.14718233816424029091331188498, −9.280216995608991192706314837188, −6.03856530151625748724604981259, −3.40835079935624569015258129348,
3.40835079935624569015258129348, 6.03856530151625748724604981259, 9.280216995608991192706314837188, 12.14718233816424029091331188498, 13.74343807720023324069913020571, 14.77087689827306271913144446404, 16.91637363899346243655810978992, 19.17096478842570809276102939166, 20.75174794706359464723370264148, 22.21199405941046639495055273162
