| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 2·6-s + 7-s − 2·9-s − 5·11-s + 2·12-s − 3·13-s − 2·14-s − 4·16-s + 5·17-s + 4·18-s + 21-s + 10·22-s + 23-s + 6·26-s − 5·27-s + 2·28-s + 3·29-s + 6·31-s + 8·32-s − 5·33-s − 10·34-s − 4·36-s + 4·37-s − 3·39-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s + 0.377·7-s − 2/3·9-s − 1.50·11-s + 0.577·12-s − 0.832·13-s − 0.534·14-s − 16-s + 1.21·17-s + 0.942·18-s + 0.218·21-s + 2.13·22-s + 0.208·23-s + 1.17·26-s − 0.962·27-s + 0.377·28-s + 0.557·29-s + 1.07·31-s + 1.41·32-s − 0.870·33-s − 1.71·34-s − 2/3·36-s + 0.657·37-s − 0.480·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.176384922727208447230330582813, −7.63807370154589821455136394048, −7.16364310068840225553097410474, −5.87569790442897986888279234516, −5.18979928366786942928980270321, −4.26683137334565487932389416652, −2.78772473421367586501771496107, −2.54579250009981993592100560124, −1.20742621489710710952356534454, 0,
1.20742621489710710952356534454, 2.54579250009981993592100560124, 2.78772473421367586501771496107, 4.26683137334565487932389416652, 5.18979928366786942928980270321, 5.87569790442897986888279234516, 7.16364310068840225553097410474, 7.63807370154589821455136394048, 8.176384922727208447230330582813
