| L(s) = 1 | + 31.3·2-s + 228.·3-s + 473.·4-s − 625·5-s + 7.17e3·6-s − 1.21e3·8-s + 3.25e4·9-s − 1.96e4·10-s − 9.01e4·11-s + 1.08e5·12-s − 1.31e5·13-s − 1.42e5·15-s − 2.80e5·16-s + 3.68e5·17-s + 1.02e6·18-s − 272.·19-s − 2.95e5·20-s − 2.82e6·22-s + 8.32e5·23-s − 2.76e5·24-s + 3.90e5·25-s − 4.12e6·26-s + 2.93e6·27-s − 4.44e6·29-s − 4.48e6·30-s − 7.48e6·31-s − 8.18e6·32-s + ⋯ |
| L(s) = 1 | + 1.38·2-s + 1.62·3-s + 0.924·4-s − 0.447·5-s + 2.25·6-s − 0.104·8-s + 1.65·9-s − 0.620·10-s − 1.85·11-s + 1.50·12-s − 1.27·13-s − 0.728·15-s − 1.06·16-s + 1.07·17-s + 2.29·18-s − 0.000479·19-s − 0.413·20-s − 2.57·22-s + 0.620·23-s − 0.170·24-s + 0.200·25-s − 1.76·26-s + 1.06·27-s − 1.16·29-s − 1.01·30-s − 1.45·31-s − 1.37·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 625T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 31.3T + 512T^{2} \) |
| 3 | \( 1 - 228.T + 1.96e4T^{2} \) |
| 11 | \( 1 + 9.01e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.31e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.68e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 272.T + 3.22e11T^{2} \) |
| 23 | \( 1 - 8.32e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.44e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.48e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 3.64e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.71e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.50e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.74e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.44e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.19e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.05e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.11e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.51e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.35e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.80e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.86e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 7.15e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.00e9T + 7.60e17T^{2} \) |
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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
−9.903522611479031166602235539166, −8.968608128376315265653635777194, −7.71190937943924344836519539167, −7.38289853802031065975814903920, −5.53967434488163939257638447452, −4.73072051186894981197440083161, −3.55186107577350580585713930181, −2.90216980601680388561737478208, −2.10755802700227672838656431860, 0,
2.10755802700227672838656431860, 2.90216980601680388561737478208, 3.55186107577350580585713930181, 4.73072051186894981197440083161, 5.53967434488163939257638447452, 7.38289853802031065975814903920, 7.71190937943924344836519539167, 8.968608128376315265653635777194, 9.903522611479031166602235539166
