| L(s) = 1 | − 1.70·2-s + 0.687·3-s + 0.895·4-s + 5-s − 1.16·6-s − 0.626·7-s + 1.87·8-s − 2.52·9-s − 1.70·10-s − 4.78·11-s + 0.615·12-s + 4.07·13-s + 1.06·14-s + 0.687·15-s − 4.98·16-s + 17-s + 4.30·18-s + 7.76·19-s + 0.895·20-s − 0.430·21-s + 8.13·22-s − 6.10·23-s + 1.29·24-s + 25-s − 6.93·26-s − 3.79·27-s − 0.561·28-s + ⋯ |
| L(s) = 1 | − 1.20·2-s + 0.396·3-s + 0.447·4-s + 0.447·5-s − 0.477·6-s − 0.236·7-s + 0.664·8-s − 0.842·9-s − 0.538·10-s − 1.44·11-s + 0.177·12-s + 1.13·13-s + 0.284·14-s + 0.177·15-s − 1.24·16-s + 0.242·17-s + 1.01·18-s + 1.78·19-s + 0.200·20-s − 0.0939·21-s + 1.73·22-s − 1.27·23-s + 0.263·24-s + 0.200·25-s − 1.35·26-s − 0.731·27-s − 0.106·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| good | 2 | \( 1 + 1.70T + 2T^{2} \) |
| 3 | \( 1 - 0.687T + 3T^{2} \) |
| 7 | \( 1 + 0.626T + 7T^{2} \) |
| 11 | \( 1 + 4.78T + 11T^{2} \) |
| 13 | \( 1 - 4.07T + 13T^{2} \) |
| 19 | \( 1 - 7.76T + 19T^{2} \) |
| 23 | \( 1 + 6.10T + 23T^{2} \) |
| 29 | \( 1 - 3.22T + 29T^{2} \) |
| 31 | \( 1 + 2.33T + 31T^{2} \) |
| 37 | \( 1 + 7.97T + 37T^{2} \) |
| 41 | \( 1 - 8.41T + 41T^{2} \) |
| 43 | \( 1 + 0.126T + 43T^{2} \) |
| 47 | \( 1 + 4.84T + 47T^{2} \) |
| 53 | \( 1 - 8.59T + 53T^{2} \) |
| 59 | \( 1 - 8.35T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 - 7.72T + 67T^{2} \) |
| 73 | \( 1 + 8.25T + 73T^{2} \) |
| 79 | \( 1 - 6.32T + 79T^{2} \) |
| 83 | \( 1 - 4.28T + 83T^{2} \) |
| 89 | \( 1 - 3.34T + 89T^{2} \) |
| 97 | \( 1 - 7.19T + 97T^{2} \) |
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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
−7.911967363236892208957313811942, −7.42652213639337271074821302791, −6.39658522941645423032230852846, −5.58857814485418329611972771735, −5.08498960573426594975334430217, −3.80727559372677939793235748287, −3.00564028141873137273270990212, −2.18384158575648215172754763193, −1.15915654386232381583544817249, 0,
1.15915654386232381583544817249, 2.18384158575648215172754763193, 3.00564028141873137273270990212, 3.80727559372677939793235748287, 5.08498960573426594975334430217, 5.58857814485418329611972771735, 6.39658522941645423032230852846, 7.42652213639337271074821302791, 7.911967363236892208957313811942
