| L(s) = 1 | + 21.3·2-s − 195.·3-s − 57.5·4-s + 1.27e3·5-s − 4.15e3·6-s + 2.27e3·7-s − 1.21e4·8-s + 1.83e4·9-s + 2.72e4·10-s + 7.17e3·11-s + 1.12e4·12-s + 4.85e4·14-s − 2.49e5·15-s − 2.29e5·16-s − 4.47e5·17-s + 3.91e5·18-s + 5.28e5·19-s − 7.35e4·20-s − 4.44e5·21-s + 1.53e5·22-s + 2.24e6·23-s + 2.36e6·24-s − 3.22e5·25-s + 2.54e5·27-s − 1.31e5·28-s + 5.98e6·29-s − 5.31e6·30-s + ⋯ |
| L(s) = 1 | + 0.942·2-s − 1.39·3-s − 0.112·4-s + 0.913·5-s − 1.31·6-s + 0.358·7-s − 1.04·8-s + 0.933·9-s + 0.860·10-s + 0.147·11-s + 0.156·12-s + 0.337·14-s − 1.27·15-s − 0.874·16-s − 1.30·17-s + 0.879·18-s + 0.930·19-s − 0.102·20-s − 0.498·21-s + 0.139·22-s + 1.67·23-s + 1.45·24-s − 0.164·25-s + 0.0921·27-s − 0.0403·28-s + 1.57·29-s − 1.19·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| good | 2 | \( 1 - 21.3T + 512T^{2} \) |
| 3 | \( 1 + 195.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.27e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 2.27e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.17e3T + 2.35e9T^{2} \) |
| 17 | \( 1 + 4.47e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.28e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.24e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.98e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.69e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.26e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.76e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.27e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.32e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.18e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.14e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 7.80e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.40e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.25e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.88e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.28e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.43e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.92e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.14e9T + 7.60e17T^{2} \) |
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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
−10.92751742752792356026109024582, −9.754625057452801321089581189823, −8.726170750207524685645644148027, −6.83113213990105204114716702012, −6.07791989853218377200619046186, −5.13990259771262711706660689510, −4.60049213204000957248942268101, −2.93046631033739325571174468314, −1.30556712586194418739685089718, 0,
1.30556712586194418739685089718, 2.93046631033739325571174468314, 4.60049213204000957248942268101, 5.13990259771262711706660689510, 6.07791989853218377200619046186, 6.83113213990105204114716702012, 8.726170750207524685645644148027, 9.754625057452801321089581189823, 10.92751742752792356026109024582
