| L(s) = 1 | − 3.53·2-s − 13.5·3-s − 19.5·4-s + 25·5-s + 48·6-s + 181.·8-s − 58.2·9-s − 88.2·10-s − 691.·11-s + 265.·12-s + 502.·13-s − 339.·15-s − 17.5·16-s + 991.·17-s + 205.·18-s − 661.·19-s − 488.·20-s + 2.44e3·22-s + 3.41e3·23-s − 2.47e3·24-s + 625·25-s − 1.77e3·26-s + 4.09e3·27-s + 6.75e3·29-s + 1.20e3·30-s + 3.92e3·31-s − 5.76e3·32-s + ⋯ |
| L(s) = 1 | − 0.624·2-s − 0.872·3-s − 0.610·4-s + 0.447·5-s + 0.544·6-s + 1.00·8-s − 0.239·9-s − 0.279·10-s − 1.72·11-s + 0.532·12-s + 0.824·13-s − 0.389·15-s − 0.0171·16-s + 0.831·17-s + 0.149·18-s − 0.420·19-s − 0.272·20-s + 1.07·22-s + 1.34·23-s − 0.876·24-s + 0.200·25-s − 0.514·26-s + 1.08·27-s + 1.49·29-s + 0.243·30-s + 0.733·31-s − 0.994·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - 25T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3.53T + 32T^{2} \) |
| 3 | \( 1 + 13.5T + 243T^{2} \) |
| 11 | \( 1 + 691.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 502.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 991.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 661.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.41e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.75e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.92e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 627.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.62e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.72e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 4.29e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.59e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 8.90e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.89e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.25e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.89e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.01e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.69e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.07e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.24e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.04e5T + 8.58e9T^{2} \) |
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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
−10.46980060444792679932791795511, −10.09595769944624293901500445824, −8.679938574054879367287235782787, −8.081125606685575868260148092923, −6.66435862089440912418136022170, −5.44120188345495381305532328127, −4.84688985956393295273805530025, −2.98667903040177048459845321610, −1.13917759008544934626375582867, 0,
1.13917759008544934626375582867, 2.98667903040177048459845321610, 4.84688985956393295273805530025, 5.44120188345495381305532328127, 6.66435862089440912418136022170, 8.081125606685575868260148092923, 8.679938574054879367287235782787, 10.09595769944624293901500445824, 10.46980060444792679932791795511
