| L(s) = 1 | + 0.894·2-s + 3-s − 1.20·4-s − 5-s + 0.894·6-s − 1.23·7-s − 2.86·8-s + 9-s − 0.894·10-s − 2.33·11-s − 1.20·12-s + 2.73·13-s − 1.10·14-s − 15-s − 0.158·16-s + 2.62·17-s + 0.894·18-s + 6.44·19-s + 1.20·20-s − 1.23·21-s − 2.09·22-s + 0.252·23-s − 2.86·24-s + 25-s + 2.44·26-s + 27-s + 1.48·28-s + ⋯ |
| L(s) = 1 | + 0.632·2-s + 0.577·3-s − 0.600·4-s − 0.447·5-s + 0.365·6-s − 0.468·7-s − 1.01·8-s + 0.333·9-s − 0.282·10-s − 0.704·11-s − 0.346·12-s + 0.758·13-s − 0.295·14-s − 0.258·15-s − 0.0395·16-s + 0.637·17-s + 0.210·18-s + 1.47·19-s + 0.268·20-s − 0.270·21-s − 0.445·22-s + 0.0526·23-s − 0.584·24-s + 0.200·25-s + 0.479·26-s + 0.192·27-s + 0.280·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 0.894T + 2T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 + 2.33T + 11T^{2} \) |
| 13 | \( 1 - 2.73T + 13T^{2} \) |
| 17 | \( 1 - 2.62T + 17T^{2} \) |
| 19 | \( 1 - 6.44T + 19T^{2} \) |
| 23 | \( 1 - 0.252T + 23T^{2} \) |
| 29 | \( 1 + 4.04T + 29T^{2} \) |
| 31 | \( 1 + 3.99T + 31T^{2} \) |
| 37 | \( 1 + 5.94T + 37T^{2} \) |
| 41 | \( 1 + 7.32T + 41T^{2} \) |
| 43 | \( 1 + 6.68T + 43T^{2} \) |
| 47 | \( 1 - 9.58T + 47T^{2} \) |
| 53 | \( 1 - 7.13T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 - 8.58T + 61T^{2} \) |
| 67 | \( 1 - 2.37T + 67T^{2} \) |
| 71 | \( 1 + 8.87T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 - 5.00T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 + 0.511T + 89T^{2} \) |
| 97 | \( 1 + 8.25T + 97T^{2} \) |
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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
−7.70158148440049128381629537925, −7.15245635201608413806217346787, −6.13908682580662072202862663438, −5.38502718020327849380955122147, −4.90003267251095460117860438999, −3.67148126815263901053583757962, −3.56157354827409294971320071167, −2.72284974691222631785560959303, −1.33085180768542065325588865019, 0,
1.33085180768542065325588865019, 2.72284974691222631785560959303, 3.56157354827409294971320071167, 3.67148126815263901053583757962, 4.90003267251095460117860438999, 5.38502718020327849380955122147, 6.13908682580662072202862663438, 7.15245635201608413806217346787, 7.70158148440049128381629537925
