| L(s) = 1 | − 9.91·2-s + 15.6·3-s + 66.2·4-s + 25·5-s − 155.·6-s − 122.·7-s − 339.·8-s + 2.59·9-s − 247.·10-s + 121·11-s + 1.03e3·12-s + 368.·13-s + 1.21e3·14-s + 391.·15-s + 1.24e3·16-s − 858.·17-s − 25.6·18-s − 361·19-s + 1.65e3·20-s − 1.91e3·21-s − 1.19e3·22-s + 1.75e3·23-s − 5.32e3·24-s + 625·25-s − 3.65e3·26-s − 3.76e3·27-s − 8.09e3·28-s + ⋯ |
| L(s) = 1 | − 1.75·2-s + 1.00·3-s + 2.07·4-s + 0.447·5-s − 1.76·6-s − 0.942·7-s − 1.87·8-s + 0.0106·9-s − 0.783·10-s + 0.301·11-s + 2.08·12-s + 0.605·13-s + 1.65·14-s + 0.449·15-s + 1.21·16-s − 0.720·17-s − 0.0186·18-s − 0.229·19-s + 0.926·20-s − 0.947·21-s − 0.528·22-s + 0.693·23-s − 1.88·24-s + 0.200·25-s − 1.06·26-s − 0.994·27-s − 1.95·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 + 9.91T + 32T^{2} \) |
| 3 | \( 1 - 15.6T + 243T^{2} \) |
| 7 | \( 1 + 122.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 368.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 858.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.75e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.39e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.18e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.43e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.96e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.92e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.35e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.74e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.22e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.21e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.44e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.14e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.79e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.69e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.32e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.01e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.21e4T + 8.58e9T^{2} \) |
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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
−9.027845980816499646679133630361, −8.198521561946988041478061593829, −7.42658714326756621525174628248, −6.55480844625666399114744435328, −5.84505418028688876124310449455, −3.99369972584991476839749777742, −2.87049141302508250469422262575, −2.22442392883789794911732665457, −1.11875823930885365712112697097, 0,
1.11875823930885365712112697097, 2.22442392883789794911732665457, 2.87049141302508250469422262575, 3.99369972584991476839749777742, 5.84505418028688876124310449455, 6.55480844625666399114744435328, 7.42658714326756621525174628248, 8.198521561946988041478061593829, 9.027845980816499646679133630361
