| L(s) = 1 | − 2·5-s + 24·7-s − 44·11-s − 22·13-s − 50·17-s − 44·19-s + 56·23-s − 121·25-s + 198·29-s − 160·31-s − 48·35-s + 162·37-s + 198·41-s − 52·43-s − 528·47-s + 233·49-s − 242·53-s + 88·55-s − 668·59-s − 550·61-s + 44·65-s − 188·67-s − 728·71-s + 154·73-s − 1.05e3·77-s − 656·79-s + 236·83-s + ⋯ |
| L(s) = 1 | − 0.178·5-s + 1.29·7-s − 1.20·11-s − 0.469·13-s − 0.713·17-s − 0.531·19-s + 0.507·23-s − 0.967·25-s + 1.26·29-s − 0.926·31-s − 0.231·35-s + 0.719·37-s + 0.754·41-s − 0.184·43-s − 1.63·47-s + 0.679·49-s − 0.627·53-s + 0.215·55-s − 1.47·59-s − 1.15·61-s + 0.0839·65-s − 0.342·67-s − 1.21·71-s + 0.246·73-s − 1.56·77-s − 0.934·79-s + 0.312·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p^{3} T^{2} \) |
| 7 | \( 1 - 24 T + p^{3} T^{2} \) |
| 11 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 + 22 T + p^{3} T^{2} \) |
| 17 | \( 1 + 50 T + p^{3} T^{2} \) |
| 19 | \( 1 + 44 T + p^{3} T^{2} \) |
| 23 | \( 1 - 56 T + p^{3} T^{2} \) |
| 29 | \( 1 - 198 T + p^{3} T^{2} \) |
| 31 | \( 1 + 160 T + p^{3} T^{2} \) |
| 37 | \( 1 - 162 T + p^{3} T^{2} \) |
| 41 | \( 1 - 198 T + p^{3} T^{2} \) |
| 43 | \( 1 + 52 T + p^{3} T^{2} \) |
| 47 | \( 1 + 528 T + p^{3} T^{2} \) |
| 53 | \( 1 + 242 T + p^{3} T^{2} \) |
| 59 | \( 1 + 668 T + p^{3} T^{2} \) |
| 61 | \( 1 + 550 T + p^{3} T^{2} \) |
| 67 | \( 1 + 188 T + p^{3} T^{2} \) |
| 71 | \( 1 + 728 T + p^{3} T^{2} \) |
| 73 | \( 1 - 154 T + p^{3} T^{2} \) |
| 79 | \( 1 + 656 T + p^{3} T^{2} \) |
| 83 | \( 1 - 236 T + p^{3} T^{2} \) |
| 89 | \( 1 + 714 T + p^{3} T^{2} \) |
| 97 | \( 1 + 478 T + p^{3} T^{2} \) |
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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
−9.994572903177907165056558540392, −8.859776279037332237775906751291, −8.005383595096572461792435259923, −7.45802210346807049811398576422, −6.16310499312910173216560955567, −5.01068821799152630351142567551, −4.43025013756735734434050265774, −2.82937782598213268909013896264, −1.71660969680678053474033781070, 0,
1.71660969680678053474033781070, 2.82937782598213268909013896264, 4.43025013756735734434050265774, 5.01068821799152630351142567551, 6.16310499312910173216560955567, 7.45802210346807049811398576422, 8.005383595096572461792435259923, 8.859776279037332237775906751291, 9.994572903177907165056558540392
