| L(s) = 1 | + 3-s + 4·5-s + 4·7-s − 2·9-s − 5·11-s − 4·13-s + 4·15-s − 2·17-s + 4·21-s + 11·25-s − 5·27-s + 4·29-s − 8·31-s − 5·33-s + 16·35-s + 8·37-s − 4·39-s − 3·41-s − 8·43-s − 8·45-s + 12·47-s + 9·49-s − 2·51-s + 12·53-s − 20·55-s + 3·59-s − 4·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s + 1.51·7-s − 2/3·9-s − 1.50·11-s − 1.10·13-s + 1.03·15-s − 0.485·17-s + 0.872·21-s + 11/5·25-s − 0.962·27-s + 0.742·29-s − 1.43·31-s − 0.870·33-s + 2.70·35-s + 1.31·37-s − 0.640·39-s − 0.468·41-s − 1.21·43-s − 1.19·45-s + 1.75·47-s + 9/7·49-s − 0.280·51-s + 1.64·53-s − 2.69·55-s + 0.390·59-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.179632422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.179632422\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−13.88812400698086, −13.51042155210599, −12.95917078385774, −12.59912446091669, −11.76898917770028, −11.35926521464443, −10.69545400631175, −10.37707117463811, −9.933465869919265, −9.258430387872911, −8.887488754864378, −8.366395522625155, −7.860032940337010, −7.385782207041435, −6.784626379892143, −5.959740558783433, −5.470914306746705, −5.183171142355749, −4.751550332642486, −3.945064966233212, −2.825678228150441, −2.559064131329364, −2.110901338458101, −1.610722830863535, −0.5767605657238609,
0.5767605657238609, 1.610722830863535, 2.110901338458101, 2.559064131329364, 2.825678228150441, 3.945064966233212, 4.751550332642486, 5.183171142355749, 5.470914306746705, 5.959740558783433, 6.784626379892143, 7.385782207041435, 7.860032940337010, 8.366395522625155, 8.887488754864378, 9.258430387872911, 9.933465869919265, 10.37707117463811, 10.69545400631175, 11.35926521464443, 11.76898917770028, 12.59912446091669, 12.95917078385774, 13.51042155210599, 13.88812400698086
