| L(s) = 1 | − 2·2-s + 2·4-s − 5-s − 7-s + 2·10-s + 6·11-s + 13-s + 2·14-s − 4·16-s − 6·19-s − 2·20-s − 12·22-s − 5·23-s + 25-s − 2·26-s − 2·28-s + 6·29-s − 9·31-s + 8·32-s + 35-s + 5·37-s + 12·38-s − 9·41-s + 12·44-s + 10·46-s + 47-s + 49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.447·5-s − 0.377·7-s + 0.632·10-s + 1.80·11-s + 0.277·13-s + 0.534·14-s − 16-s − 1.37·19-s − 0.447·20-s − 2.55·22-s − 1.04·23-s + 1/5·25-s − 0.392·26-s − 0.377·28-s + 1.11·29-s − 1.61·31-s + 1.41·32-s + 0.169·35-s + 0.821·37-s + 1.94·38-s − 1.40·41-s + 1.80·44-s + 1.47·46-s + 0.145·47-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
| \(L(1)\) |
\(\approx\) |
\(0.9171731177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9171731177\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−13.91579573452022, −13.40037981122057, −12.63268535894289, −12.36027826510225, −11.62499579073200, −11.34257322557359, −10.83631523561203, −10.22653226177301, −9.794645127270136, −9.364122736092247, −8.795291631494213, −8.373742213792349, −8.155879196528791, −7.239613545082380, −6.830344790533544, −6.508531564572054, −5.911935827196667, −5.056361919382256, −4.192030166069263, −3.973687371153236, −3.350983726978118, −2.235367589443711, −1.897975627912417, −1.028980290296682, −0.4616273191867133,
0.4616273191867133, 1.028980290296682, 1.897975627912417, 2.235367589443711, 3.350983726978118, 3.973687371153236, 4.192030166069263, 5.056361919382256, 5.911935827196667, 6.508531564572054, 6.830344790533544, 7.239613545082380, 8.155879196528791, 8.373742213792349, 8.795291631494213, 9.364122736092247, 9.794645127270136, 10.22653226177301, 10.83631523561203, 11.34257322557359, 11.62499579073200, 12.36027826510225, 12.63268535894289, 13.40037981122057, 13.91579573452022
