| L(s) = 1 | + 2.72e3·2-s − 5.08e4·3-s + 5.34e6·4-s − 1.38e8·6-s − 3.19e8·7-s + 8.85e9·8-s − 7.87e9·9-s − 1.64e11·11-s − 2.71e11·12-s + 5.81e11·13-s − 8.71e11·14-s + 1.29e13·16-s − 3.89e12·17-s − 2.14e13·18-s − 3.65e13·19-s + 1.62e13·21-s − 4.48e14·22-s − 2.08e14·23-s − 4.49e14·24-s + 1.58e15·26-s + 9.32e14·27-s − 1.70e15·28-s + 8.16e14·29-s − 5.45e15·31-s + 1.67e16·32-s + 8.36e15·33-s − 1.06e16·34-s + ⋯ |
| L(s) = 1 | + 1.88·2-s − 0.497·3-s + 2.54·4-s − 0.936·6-s − 0.427·7-s + 2.91·8-s − 0.752·9-s − 1.91·11-s − 1.26·12-s + 1.17·13-s − 0.805·14-s + 2.94·16-s − 0.468·17-s − 1.41·18-s − 1.36·19-s + 0.212·21-s − 3.60·22-s − 1.04·23-s − 1.44·24-s + 2.20·26-s + 0.871·27-s − 1.08·28-s + 0.360·29-s − 1.19·31-s + 2.62·32-s + 0.950·33-s − 0.883·34-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 | \( 1 - 2.72e3T + 2.09e6T^{2} \) |
| 3 | \( 1 + 5.08e4T + 1.04e10T^{2} \) |
| 7 | \( 1 + 3.19e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 1.64e11T + 7.40e21T^{2} \) |
| 13 | \( 1 - 5.81e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 3.89e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 3.65e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 2.08e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 8.16e14T + 5.13e30T^{2} \) |
| 31 | \( 1 + 5.45e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 3.79e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 9.62e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 6.14e16T + 2.00e34T^{2} \) |
| 47 | \( 1 + 4.95e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 9.65e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 3.38e16T + 1.54e37T^{2} \) |
| 61 | \( 1 - 1.46e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.95e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 6.92e18T + 7.52e38T^{2} \) |
| 73 | \( 1 - 2.18e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 6.04e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 1.33e19T + 1.99e40T^{2} \) |
| 89 | \( 1 + 8.06e19T + 8.65e40T^{2} \) |
| 97 | \( 1 - 7.05e20T + 5.27e41T^{2} \) |
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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
−12.74990519400682813282936260228, −11.30181338417551742239677592583, −10.61131494400656173079599216667, −8.051471759533591863025652477931, −6.37137026255345502770883791356, −5.70174395234088670013946091918, −4.52796093864458518408790881084, −3.18983033052418795333470443062, −2.15878088663674426566110250490, 0,
2.15878088663674426566110250490, 3.18983033052418795333470443062, 4.52796093864458518408790881084, 5.70174395234088670013946091918, 6.37137026255345502770883791356, 8.051471759533591863025652477931, 10.61131494400656173079599216667, 11.30181338417551742239677592583, 12.74990519400682813282936260228
