| L(s) = 1 | − 2.68·2-s + 2.77·3-s + 5.19·4-s − 2.34·5-s − 7.44·6-s − 2.63·7-s − 8.56·8-s + 4.70·9-s + 6.29·10-s − 4.33·11-s + 14.4·12-s + 1.38·13-s + 7.07·14-s − 6.51·15-s + 12.5·16-s + 3.49·17-s − 12.6·18-s + 6.49·19-s − 12.1·20-s − 7.32·21-s + 11.6·22-s − 5.61·23-s − 23.7·24-s + 0.503·25-s − 3.71·26-s + 4.74·27-s − 13.6·28-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 1.60·3-s + 2.59·4-s − 1.04·5-s − 3.03·6-s − 0.996·7-s − 3.02·8-s + 1.56·9-s + 1.98·10-s − 1.30·11-s + 4.16·12-s + 0.384·13-s + 1.89·14-s − 1.68·15-s + 3.14·16-s + 0.846·17-s − 2.97·18-s + 1.49·19-s − 2.72·20-s − 1.59·21-s + 2.47·22-s − 1.17·23-s − 4.85·24-s + 0.100·25-s − 0.728·26-s + 0.912·27-s − 2.58·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 787 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 787 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 787 | \( 1 + T \) |
| good | 2 | \( 1 + 2.68T + 2T^{2} \) |
| 3 | \( 1 - 2.77T + 3T^{2} \) |
| 5 | \( 1 + 2.34T + 5T^{2} \) |
| 7 | \( 1 + 2.63T + 7T^{2} \) |
| 11 | \( 1 + 4.33T + 11T^{2} \) |
| 13 | \( 1 - 1.38T + 13T^{2} \) |
| 17 | \( 1 - 3.49T + 17T^{2} \) |
| 19 | \( 1 - 6.49T + 19T^{2} \) |
| 23 | \( 1 + 5.61T + 23T^{2} \) |
| 29 | \( 1 + 5.82T + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 + 6.59T + 37T^{2} \) |
| 41 | \( 1 - 6.49T + 41T^{2} \) |
| 43 | \( 1 + 1.41T + 43T^{2} \) |
| 47 | \( 1 + 1.45T + 47T^{2} \) |
| 53 | \( 1 + 9.68T + 53T^{2} \) |
| 59 | \( 1 + 1.63T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 - 0.376T + 67T^{2} \) |
| 71 | \( 1 - 3.82T + 71T^{2} \) |
| 73 | \( 1 + 15.9T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 - 4.84T + 83T^{2} \) |
| 89 | \( 1 + 3.58T + 89T^{2} \) |
| 97 | \( 1 + 6.16T + 97T^{2} \) |
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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
−9.554677991146778326124530672047, −9.093899795722718730658195648690, −8.043230518442626873535814639117, −7.70761230047439285758573193602, −7.23967073924776755110450587888, −5.78837299517788410154496985924, −3.51512956497922017701340073365, −3.12388369272939284321701346821, −1.84726531268174961207186058637, 0,
1.84726531268174961207186058637, 3.12388369272939284321701346821, 3.51512956497922017701340073365, 5.78837299517788410154496985924, 7.23967073924776755110450587888, 7.70761230047439285758573193602, 8.043230518442626873535814639117, 9.093899795722718730658195648690, 9.554677991146778326124530672047
