| L(s) = 1 | + 2-s − 4-s + 5-s + 7-s − 3·8-s + 10-s + 4·11-s + 6·13-s + 14-s − 16-s − 4·19-s − 20-s + 4·22-s + 25-s + 6·26-s − 28-s + 6·29-s + 8·31-s + 5·32-s + 35-s + 2·37-s − 4·38-s − 3·40-s − 6·41-s + 4·43-s − 4·44-s + 49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s + 0.377·7-s − 1.06·8-s + 0.316·10-s + 1.20·11-s + 1.66·13-s + 0.267·14-s − 1/4·16-s − 0.917·19-s − 0.223·20-s + 0.852·22-s + 1/5·25-s + 1.17·26-s − 0.188·28-s + 1.11·29-s + 1.43·31-s + 0.883·32-s + 0.169·35-s + 0.328·37-s − 0.648·38-s − 0.474·40-s − 0.937·41-s + 0.609·43-s − 0.603·44-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.918704112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.918704112\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.80998926370910, −13.45412611959736, −13.05229111947515, −12.40073016071822, −11.98179726134958, −11.51730303181213, −10.99319983749466, −10.37591445702900, −9.914534493528347, −9.282790983119570, −8.718237855616954, −8.515563806045868, −8.043367355284452, −6.992958984642166, −6.532513864163987, −6.023003948611324, −5.827490701373834, −4.819123631639119, −4.578190142625972, −3.901612870311356, −3.526794660224373, −2.790686055770986, −2.026205383401967, −1.191775333338579, −0.7267353138406618,
0.7267353138406618, 1.191775333338579, 2.026205383401967, 2.790686055770986, 3.526794660224373, 3.901612870311356, 4.578190142625972, 4.819123631639119, 5.827490701373834, 6.023003948611324, 6.532513864163987, 6.992958984642166, 8.043367355284452, 8.515563806045868, 8.718237855616954, 9.282790983119570, 9.914534493528347, 10.37591445702900, 10.99319983749466, 11.51730303181213, 11.98179726134958, 12.40073016071822, 13.05229111947515, 13.45412611959736, 13.80998926370910
