| L(s) = 1 | + 22.5·2-s + 66.2·3-s − 3.88·4-s − 391.·5-s + 1.49e3·6-s + 2.40e3·7-s − 1.16e4·8-s − 1.52e4·9-s − 8.82e3·10-s + 9.35e4·11-s − 257.·12-s + 2.85e4·13-s + 5.41e4·14-s − 2.59e4·15-s − 2.60e5·16-s − 3.85e5·17-s − 3.44e5·18-s − 1.01e6·19-s + 1.52e3·20-s + 1.59e5·21-s + 2.10e6·22-s + 1.82e6·23-s − 7.70e5·24-s − 1.79e6·25-s + 6.43e5·26-s − 2.31e6·27-s − 9.33e3·28-s + ⋯ |
| L(s) = 1 | + 0.996·2-s + 0.472·3-s − 0.00759·4-s − 0.280·5-s + 0.470·6-s + 0.377·7-s − 1.00·8-s − 0.776·9-s − 0.279·10-s + 1.92·11-s − 0.00358·12-s + 0.277·13-s + 0.376·14-s − 0.132·15-s − 0.992·16-s − 1.12·17-s − 0.773·18-s − 1.79·19-s + 0.00212·20-s + 0.178·21-s + 1.92·22-s + 1.36·23-s − 0.474·24-s − 0.921·25-s + 0.276·26-s − 0.839·27-s − 0.00286·28-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 | \( 1 - 22.5T + 512T^{2} \) |
| 3 | \( 1 - 66.2T + 1.96e4T^{2} \) |
| 5 | \( 1 + 391.T + 1.95e6T^{2} \) |
| 11 | \( 1 - 9.35e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 3.85e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.01e6T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.82e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.33e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.13e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.31e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 7.87e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.05e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.94e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.09e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 6.16e5T + 8.66e15T^{2} \) |
| 61 | \( 1 - 5.11e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.58e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 9.17e6T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.70e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.66e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.26e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.11e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 4.95e8T + 7.60e17T^{2} \) |
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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
−11.81897730468975660596436318807, −11.08537308256316737364975083070, −8.982700026812559073774346044350, −8.788780272687811153590512037899, −6.87910067643945071666998637259, −5.75656203817492236062474025113, −4.28233804387866505862941837557, −3.57823833065120818841291405005, −1.97900869221996322202415656684, 0,
1.97900869221996322202415656684, 3.57823833065120818841291405005, 4.28233804387866505862941837557, 5.75656203817492236062474025113, 6.87910067643945071666998637259, 8.788780272687811153590512037899, 8.982700026812559073774346044350, 11.08537308256316737364975083070, 11.81897730468975660596436318807
