L(s) = 1 | + 5-s + 11-s − 3·13-s + 2·17-s − 8·19-s − 6·23-s + 25-s + 6·29-s + 9·31-s + 8·37-s − 8·41-s + 7·43-s + 4·47-s + 2·53-s + 55-s − 5·59-s + 14·61-s − 3·65-s + 4·67-s + 9·71-s − 73-s + 12·79-s + 9·83-s + 2·85-s + 7·89-s − 8·95-s + 6·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.301·11-s − 0.832·13-s + 0.485·17-s − 1.83·19-s − 1.25·23-s + 1/5·25-s + 1.11·29-s + 1.61·31-s + 1.31·37-s − 1.24·41-s + 1.06·43-s + 0.583·47-s + 0.274·53-s + 0.134·55-s − 0.650·59-s + 1.79·61-s − 0.372·65-s + 0.488·67-s + 1.06·71-s − 0.117·73-s + 1.35·79-s + 0.987·83-s + 0.216·85-s + 0.741·89-s − 0.820·95-s + 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.122859410\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.122859410\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.55461614291980, −11.99545526759440, −11.71272030863397, −11.07161712809023, −10.49859617032288, −10.22380097557365, −9.808150112344661, −9.414413008263072, −8.766017758232442, −8.342944056533249, −7.986998878279039, −7.500689875874604, −6.761835964071409, −6.378322467510756, −6.192611432472360, −5.473748271928418, −4.945641335838138, −4.433048619883254, −4.080755190203495, −3.445097786067534, −2.672932527649429, −2.297388023779589, −1.911702333342016, −0.9541088863373516, −0.5214365634282056,
0.5214365634282056, 0.9541088863373516, 1.911702333342016, 2.297388023779589, 2.672932527649429, 3.445097786067534, 4.080755190203495, 4.433048619883254, 4.945641335838138, 5.473748271928418, 6.192611432472360, 6.378322467510756, 6.761835964071409, 7.500689875874604, 7.986998878279039, 8.342944056533249, 8.766017758232442, 9.414413008263072, 9.808150112344661, 10.22380097557365, 10.49859617032288, 11.07161712809023, 11.71272030863397, 11.99545526759440, 12.55461614291980