| L(s) = 1 | − 24.2·2-s + 6.29·3-s + 74.2·4-s + 2.01e3·5-s − 152.·6-s + 2.40e3·7-s + 1.05e4·8-s − 1.96e4·9-s − 4.87e4·10-s − 3.00e4·11-s + 467.·12-s + 2.85e4·13-s − 5.81e4·14-s + 1.26e4·15-s − 2.94e5·16-s − 5.76e5·17-s + 4.75e5·18-s + 2.82e5·19-s + 1.49e5·20-s + 1.51e4·21-s + 7.27e5·22-s + 1.75e6·23-s + 6.67e4·24-s + 2.09e6·25-s − 6.91e5·26-s − 2.47e5·27-s + 1.78e5·28-s + ⋯ |
| L(s) = 1 | − 1.07·2-s + 0.0448·3-s + 0.144·4-s + 1.43·5-s − 0.0480·6-s + 0.377·7-s + 0.914·8-s − 0.997·9-s − 1.54·10-s − 0.618·11-s + 0.00650·12-s + 0.277·13-s − 0.404·14-s + 0.0645·15-s − 1.12·16-s − 1.67·17-s + 1.06·18-s + 0.497·19-s + 0.208·20-s + 0.0169·21-s + 0.662·22-s + 1.31·23-s + 0.0410·24-s + 1.07·25-s − 0.296·26-s − 0.0896·27-s + 0.0547·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - 2.40e3T \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 2 | \( 1 + 24.2T + 512T^{2} \) |
| 3 | \( 1 - 6.29T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.01e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 3.00e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 5.76e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.82e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.75e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.87e4T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.18e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.71e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.07e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.13e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.74e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.09e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 3.11e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.96e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.53e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.89e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.45e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.55e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.55e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 8.80e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.53e8T + 7.60e17T^{2} \) |
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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
−11.17067670584652614234377462125, −10.53660515859274805443444745725, −9.197647860428733299975597444397, −8.842175286954762096304958397796, −7.42324722947667685769357165224, −5.97158632804963056191780432951, −4.83833672264541511282266434014, −2.59848549064208036613353458827, −1.48156595976440005263764823808, 0,
1.48156595976440005263764823808, 2.59848549064208036613353458827, 4.83833672264541511282266434014, 5.97158632804963056191780432951, 7.42324722947667685769357165224, 8.842175286954762096304958397796, 9.197647860428733299975597444397, 10.53660515859274805443444745725, 11.17067670584652614234377462125
