| L(s) = 1 | − 3.70·2-s − 3·3-s + 5.70·4-s + 9.10·5-s + 11.1·6-s − 5.59·7-s + 8.50·8-s + 9·9-s − 33.7·10-s + 9.29·11-s − 17.1·12-s − 77.1·13-s + 20.7·14-s − 27.3·15-s − 77.1·16-s − 61.8·17-s − 33.3·18-s − 39.3·19-s + 51.9·20-s + 16.7·21-s − 34.4·22-s − 37.4·23-s − 25.5·24-s − 42.1·25-s + 285.·26-s − 27·27-s − 31.9·28-s + ⋯ |
| L(s) = 1 | − 1.30·2-s − 0.577·3-s + 0.712·4-s + 0.814·5-s + 0.755·6-s − 0.302·7-s + 0.375·8-s + 0.333·9-s − 1.06·10-s + 0.254·11-s − 0.411·12-s − 1.64·13-s + 0.395·14-s − 0.470·15-s − 1.20·16-s − 0.881·17-s − 0.436·18-s − 0.474·19-s + 0.580·20-s + 0.174·21-s − 0.333·22-s − 0.339·23-s − 0.217·24-s − 0.336·25-s + 2.15·26-s − 0.192·27-s − 0.215·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 + 3.70T + 8T^{2} \) |
| 5 | \( 1 - 9.10T + 125T^{2} \) |
| 7 | \( 1 + 5.59T + 343T^{2} \) |
| 11 | \( 1 - 9.29T + 1.33e3T^{2} \) |
| 13 | \( 1 + 77.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 61.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 39.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 37.4T + 1.21e4T^{2} \) |
| 31 | \( 1 - 26.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 279.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 194.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 309.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 109.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 560.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 147.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 278.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 970.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 519.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 264.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 875.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 141.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 970.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 2.93T + 9.12e5T^{2} \) |
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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
−13.05838852396617139572917517950, −11.85296505754422812668227047654, −10.54317556616598078652481864878, −9.834048571652557124409324352768, −8.958724386251939969761655661281, −7.47988783527563148804368880769, −6.37771603067226241068438630885, −4.77133099791823237690241119565, −2.02029832060647282336865556047, 0,
2.02029832060647282336865556047, 4.77133099791823237690241119565, 6.37771603067226241068438630885, 7.47988783527563148804368880769, 8.958724386251939969761655661281, 9.834048571652557124409324352768, 10.54317556616598078652481864878, 11.85296505754422812668227047654, 13.05838852396617139572917517950
