| L(s) = 1 | + 11.8·2-s − 81·3-s − 370.·4-s − 236.·5-s − 963.·6-s + 4.18e3·7-s − 1.05e4·8-s + 6.56e3·9-s − 2.81e3·10-s − 2.58e4·11-s + 3.00e4·12-s + 1.09e4·13-s + 4.98e4·14-s + 1.91e4·15-s + 6.47e4·16-s − 7.48e4·17-s + 7.80e4·18-s + 5.75e5·19-s + 8.77e4·20-s − 3.39e5·21-s − 3.07e5·22-s + 1.18e6·23-s + 8.50e5·24-s − 1.89e6·25-s + 1.30e5·26-s − 5.31e5·27-s − 1.55e6·28-s + ⋯ |
| L(s) = 1 | + 0.525·2-s − 0.577·3-s − 0.723·4-s − 0.169·5-s − 0.303·6-s + 0.658·7-s − 0.906·8-s + 0.333·9-s − 0.0890·10-s − 0.531·11-s + 0.417·12-s + 0.106·13-s + 0.346·14-s + 0.0978·15-s + 0.246·16-s − 0.217·17-s + 0.175·18-s + 1.01·19-s + 0.122·20-s − 0.380·21-s − 0.279·22-s + 0.884·23-s + 0.523·24-s − 0.971·25-s + 0.0560·26-s − 0.192·27-s − 0.476·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 81T \) |
| 59 | \( 1 + 1.21e7T \) |
| good | 2 | \( 1 - 11.8T + 512T^{2} \) |
| 5 | \( 1 + 236.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 4.18e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.58e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.09e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 7.48e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.75e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.18e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 9.71e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 7.66e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.34e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.71e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.42e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.03e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.53e7T + 3.29e15T^{2} \) |
| 61 | \( 1 + 1.39e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.76e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 4.76e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.70e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 7.11e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.04e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.08e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 5.99e8T + 7.60e17T^{2} \) |
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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
−10.64372532793176872190527546431, −9.560542782628097239002629479264, −8.459360967377174663093272835514, −7.41231413876220956929696175901, −5.97540261002214412623807053004, −5.08345313880842115572227785118, −4.29192131271929068598317890488, −2.95911688963091854842478621133, −1.20294252736642915769046442990, 0,
1.20294252736642915769046442990, 2.95911688963091854842478621133, 4.29192131271929068598317890488, 5.08345313880842115572227785118, 5.97540261002214412623807053004, 7.41231413876220956929696175901, 8.459360967377174663093272835514, 9.560542782628097239002629479264, 10.64372532793176872190527546431
