| L(s) = 1 | + 5-s − 7-s − 3·9-s + 13-s + 2·17-s + 8·19-s − 4·25-s + 9·29-s − 4·31-s − 35-s − 6·37-s − 5·41-s + 8·43-s − 3·45-s + 49-s + 9·53-s + 5·61-s + 3·63-s + 65-s + 16·67-s + 12·71-s − 73-s + 4·79-s + 9·81-s + 12·83-s + 2·85-s − 9·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 9-s + 0.277·13-s + 0.485·17-s + 1.83·19-s − 4/5·25-s + 1.67·29-s − 0.718·31-s − 0.169·35-s − 0.986·37-s − 0.780·41-s + 1.21·43-s − 0.447·45-s + 1/7·49-s + 1.23·53-s + 0.640·61-s + 0.377·63-s + 0.124·65-s + 1.95·67-s + 1.42·71-s − 0.117·73-s + 0.450·79-s + 81-s + 1.31·83-s + 0.216·85-s − 0.953·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.036556546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.036556546\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−12.91880189219945, −12.31870756380420, −11.95942821636834, −11.63592380143498, −11.05099026480076, −10.56282772122090, −10.04408734430061, −9.657520323552800, −9.211877781747323, −8.746165348337484, −8.129210403197265, −7.873954614318741, −7.081968238583765, −6.783600377785821, −6.109233565849736, −5.674524855014277, −5.244709591148415, −4.930110036747563, −3.820520021768935, −3.645203328526586, −2.960214129349955, −2.502534210882245, −1.855446980126628, −1.007199596529676, −0.5558008525620920,
0.5558008525620920, 1.007199596529676, 1.855446980126628, 2.502534210882245, 2.960214129349955, 3.645203328526586, 3.820520021768935, 4.930110036747563, 5.244709591148415, 5.674524855014277, 6.109233565849736, 6.783600377785821, 7.081968238583765, 7.873954614318741, 8.129210403197265, 8.746165348337484, 9.211877781747323, 9.657520323552800, 10.04408734430061, 10.56282772122090, 11.05099026480076, 11.63592380143498, 11.95942821636834, 12.31870756380420, 12.91880189219945
