| 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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 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
−14.54786280226125, −13.71825879810202, −13.07415319876114, −12.73420110334316, −12.47579579130109, −11.68813462926076, −11.60642932469949, −10.87498264371097, −10.52800890713099, −9.940504709119366, −9.421741424320536, −8.755608687752949, −8.183472370942012, −7.822642759827724, −7.223671987241352, −6.776100006838958, −6.316340399297893, −5.433948070466648, −5.163885967612183, −4.464116130593018, −3.912101292382818, −3.110689283113708, −2.932176392004170, −2.248468260391836, −0.7470104874332543, 0, 0,
0.7470104874332543, 2.248468260391836, 2.932176392004170, 3.110689283113708, 3.912101292382818, 4.464116130593018, 5.163885967612183, 5.433948070466648, 6.316340399297893, 6.776100006838958, 7.223671987241352, 7.822642759827724, 8.183472370942012, 8.755608687752949, 9.421741424320536, 9.940504709119366, 10.52800890713099, 10.87498264371097, 11.60642932469949, 11.68813462926076, 12.47579579130109, 12.73420110334316, 13.07415319876114, 13.71825879810202, 14.54786280226125
