| L(s) = 1 | + 3-s + 5-s + 4·7-s + 9-s − 4·11-s − 2·13-s + 15-s + 2·17-s + 8·19-s + 4·21-s + 25-s + 27-s + 6·29-s − 8·31-s − 4·33-s + 4·35-s − 2·39-s − 6·41-s + 4·43-s + 45-s + 12·47-s + 9·49-s + 2·51-s − 6·53-s − 4·55-s + 8·57-s + 6·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.258·15-s + 0.485·17-s + 1.83·19-s + 0.872·21-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.696·33-s + 0.676·35-s − 0.320·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s + 1.75·47-s + 9/7·49-s + 0.280·51-s − 0.824·53-s − 0.539·55-s + 1.05·57-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.999271222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.999271222\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−12.57278355936119, −12.13697178756986, −11.76013939728114, −11.18369014966897, −10.78281363705140, −10.18029972155589, −10.03973602865119, −9.370447264832692, −8.840143504864278, −8.551998807107560, −7.790532953321127, −7.601636819132460, −7.375857619692500, −6.674173105964013, −5.802376996628135, −5.426168085290074, −5.123979721614809, −4.640627139086967, −4.070307934982718, −3.305175662485765, −2.835040967834258, −2.391428071723528, −1.711274426542006, −1.300892417369033, −0.5598126584386287,
0.5598126584386287, 1.300892417369033, 1.711274426542006, 2.391428071723528, 2.835040967834258, 3.305175662485765, 4.070307934982718, 4.640627139086967, 5.123979721614809, 5.426168085290074, 5.802376996628135, 6.674173105964013, 7.375857619692500, 7.601636819132460, 7.790532953321127, 8.551998807107560, 8.840143504864278, 9.370447264832692, 10.03973602865119, 10.18029972155589, 10.78281363705140, 11.18369014966897, 11.76013939728114, 12.13697178756986, 12.57278355936119
