| L(s) = 1 | + 4·2-s + 14.3·3-s + 16·4-s − 25·5-s + 57.2·6-s + 64·8-s − 38.5·9-s − 100·10-s + 425.·11-s + 228.·12-s − 399.·13-s − 357.·15-s + 256·16-s − 1.75e3·17-s − 154.·18-s − 2.87e3·19-s − 400·20-s + 1.70e3·22-s − 2.31e3·23-s + 915.·24-s + 625·25-s − 1.59e3·26-s − 4.02e3·27-s − 2.12e3·29-s − 1.43e3·30-s + 1.02e4·31-s + 1.02e3·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.917·3-s + 0.5·4-s − 0.447·5-s + 0.648·6-s + 0.353·8-s − 0.158·9-s − 0.316·10-s + 1.06·11-s + 0.458·12-s − 0.655·13-s − 0.410·15-s + 0.250·16-s − 1.46·17-s − 0.112·18-s − 1.82·19-s − 0.223·20-s + 0.750·22-s − 0.911·23-s + 0.324·24-s + 0.200·25-s − 0.463·26-s − 1.06·27-s − 0.469·29-s − 0.290·30-s + 1.91·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 5 | \( 1 + 25T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 14.3T + 243T^{2} \) |
| 11 | \( 1 - 425.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 399.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.75e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.87e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.31e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.12e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.02e4T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.26e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.89e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.01e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.00e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.39e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.31e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.22e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.75e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.65e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.24e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.16e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.52e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.44e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.21e4T + 8.58e9T^{2} \) |
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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
−9.586498611483172128033961205375, −8.684579154607679968169777715773, −8.020659459812178581530137588054, −6.82968662981443677591956370242, −6.12343645459934843451215947705, −4.52632041149156779985102163881, −3.99532480429640970123594059100, −2.75923186845076801243991219423, −1.90583997557873671535090861539, 0,
1.90583997557873671535090861539, 2.75923186845076801243991219423, 3.99532480429640970123594059100, 4.52632041149156779985102163881, 6.12343645459934843451215947705, 6.82968662981443677591956370242, 8.020659459812178581530137588054, 8.684579154607679968169777715773, 9.586498611483172128033961205375
