| L(s) = 1 | − 745.·2-s + 1.22e5·3-s − 1.54e6·4-s − 9.13e7·6-s − 1.12e9·7-s + 2.71e9·8-s + 4.57e9·9-s + 1.60e10·11-s − 1.88e11·12-s + 8.15e11·13-s + 8.35e11·14-s + 1.21e12·16-s + 9.20e12·17-s − 3.40e12·18-s − 1.84e13·19-s − 1.37e14·21-s − 1.19e13·22-s + 1.77e14·23-s + 3.32e14·24-s − 6.08e14·26-s − 7.22e14·27-s + 1.72e15·28-s − 2.30e15·29-s − 4.01e15·31-s − 6.59e15·32-s + 1.97e15·33-s − 6.86e15·34-s + ⋯ |
| L(s) = 1 | − 0.514·2-s + 1.19·3-s − 0.735·4-s − 0.617·6-s − 1.49·7-s + 0.893·8-s + 0.437·9-s + 0.186·11-s − 0.881·12-s + 1.64·13-s + 0.771·14-s + 0.275·16-s + 1.10·17-s − 0.224·18-s − 0.689·19-s − 1.79·21-s − 0.0962·22-s + 0.891·23-s + 1.07·24-s − 0.844·26-s − 0.674·27-s + 1.10·28-s − 1.01·29-s − 0.879·31-s − 1.03·32-s + 0.224·33-s − 0.570·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 745.T + 2.09e6T^{2} \) |
| 3 | \( 1 - 1.22e5T + 1.04e10T^{2} \) |
| 7 | \( 1 + 1.12e9T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.60e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 8.15e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 9.20e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 1.84e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 1.77e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 2.30e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 4.01e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 2.97e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 1.43e17T + 7.38e33T^{2} \) |
| 43 | \( 1 + 5.33e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 2.28e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 8.31e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 5.47e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 6.66e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 4.83e17T + 2.22e38T^{2} \) |
| 71 | \( 1 + 1.06e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 6.72e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 3.47e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.29e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 5.75e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 4.37e20T + 5.27e41T^{2} \) |
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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
−12.81878179565862612932806850231, −10.62371559448155897761196161780, −9.319756836708366985370370542558, −8.853014389632036273824332161350, −7.56857683454031767077040727650, −5.93187170934307073080236887861, −3.85655905378493122390298986946, −3.15953045113940953110265014506, −1.39075963377513101576331770829, 0,
1.39075963377513101576331770829, 3.15953045113940953110265014506, 3.85655905378493122390298986946, 5.93187170934307073080236887861, 7.56857683454031767077040727650, 8.853014389632036273824332161350, 9.319756836708366985370370542558, 10.62371559448155897761196161780, 12.81878179565862612932806850231
