| L(s) = 1 | − 16·2-s + 189.·3-s + 256·4-s − 2.46e3·5-s − 3.02e3·6-s + 2.20e3·7-s − 4.09e3·8-s + 1.60e4·9-s + 3.93e4·10-s − 1.46e4·11-s + 4.84e4·12-s − 1.40e5·13-s − 3.52e4·14-s − 4.65e5·15-s + 6.55e4·16-s − 3.43e5·17-s − 2.57e5·18-s + 1.88e5·19-s − 6.29e5·20-s + 4.16e5·21-s + 2.34e5·22-s − 2.31e6·23-s − 7.74e5·24-s + 4.10e6·25-s + 2.24e6·26-s − 6.79e5·27-s + 5.64e5·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.34·3-s + 0.5·4-s − 1.76·5-s − 0.953·6-s + 0.347·7-s − 0.353·8-s + 0.817·9-s + 1.24·10-s − 0.301·11-s + 0.674·12-s − 1.36·13-s − 0.245·14-s − 2.37·15-s + 0.250·16-s − 0.997·17-s − 0.578·18-s + 0.332·19-s − 0.880·20-s + 0.467·21-s + 0.213·22-s − 1.72·23-s − 0.476·24-s + 2.09·25-s + 0.963·26-s − 0.246·27-s + 0.173·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 16T \) |
| 11 | \( 1 + 1.46e4T \) |
| good | 3 | \( 1 - 189.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.46e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 2.20e3T + 4.03e7T^{2} \) |
| 13 | \( 1 + 1.40e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.43e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.88e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.31e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.63e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.09e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.24e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.99e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.52e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.93e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.79e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 2.28e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 4.69e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.94e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 9.62e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.14e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.72e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.19e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.24e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 4.29e5T + 7.60e17T^{2} \) |
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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
−15.26728600385056577542467715292, −14.38205586195253711362988505677, −12.43935545131287576210859686348, −11.15685180452479996982594280804, −9.363656615848476109924289887953, −8.044394305231631791141588211627, −7.50261571657642843043288788298, −4.12074172629341009915260466555, −2.49930774447671632607690690401, 0,
2.49930774447671632607690690401, 4.12074172629341009915260466555, 7.50261571657642843043288788298, 8.044394305231631791141588211627, 9.363656615848476109924289887953, 11.15685180452479996982594280804, 12.43935545131287576210859686348, 14.38205586195253711362988505677, 15.26728600385056577542467715292
