| L(s) = 1 | + 2.80·2-s + 0.554·3-s + 5.85·4-s + 1.55·6-s + 3.04·7-s + 10.7·8-s − 2.69·9-s − 2.93·11-s + 3.24·12-s + 3.24·13-s + 8.54·14-s + 18.5·16-s + 2.15·17-s − 7.54·18-s + 1.69·21-s − 8.23·22-s − 1.19·23-s + 5.98·24-s + 9.09·26-s − 3.15·27-s + 17.8·28-s + 1.77·29-s + 9.34·31-s + 30.3·32-s − 1.63·33-s + 6.04·34-s − 15.7·36-s + ⋯ |
| L(s) = 1 | + 1.98·2-s + 0.320·3-s + 2.92·4-s + 0.634·6-s + 1.15·7-s + 3.81·8-s − 0.897·9-s − 0.886·11-s + 0.937·12-s + 0.900·13-s + 2.28·14-s + 4.63·16-s + 0.523·17-s − 1.77·18-s + 0.369·21-s − 1.75·22-s − 0.249·23-s + 1.22·24-s + 1.78·26-s − 0.607·27-s + 3.37·28-s + 0.329·29-s + 1.67·31-s + 5.36·32-s − 0.283·33-s + 1.03·34-s − 2.62·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.86548024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.86548024\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.80T + 2T^{2} \) |
| 3 | \( 1 - 0.554T + 3T^{2} \) |
| 7 | \( 1 - 3.04T + 7T^{2} \) |
| 11 | \( 1 + 2.93T + 11T^{2} \) |
| 13 | \( 1 - 3.24T + 13T^{2} \) |
| 17 | \( 1 - 2.15T + 17T^{2} \) |
| 23 | \( 1 + 1.19T + 23T^{2} \) |
| 29 | \( 1 - 1.77T + 29T^{2} \) |
| 31 | \( 1 - 9.34T + 31T^{2} \) |
| 37 | \( 1 + 1.15T + 37T^{2} \) |
| 41 | \( 1 + 8.57T + 41T^{2} \) |
| 43 | \( 1 + 5.27T + 43T^{2} \) |
| 47 | \( 1 + 2.35T + 47T^{2} \) |
| 53 | \( 1 - 8.82T + 53T^{2} \) |
| 59 | \( 1 - 5.70T + 59T^{2} \) |
| 61 | \( 1 + 9.96T + 61T^{2} \) |
| 67 | \( 1 - 4.98T + 67T^{2} \) |
| 71 | \( 1 + 2.70T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 + 5.66T + 79T^{2} \) |
| 83 | \( 1 - 3.00T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 3.24T + 97T^{2} \) |
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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
−7.70203466342344005251643661101, −6.77965047928472316027355956812, −6.10334097807826589889646597817, −5.46808698406048069448159372901, −4.98605794679932717429857378222, −4.33367347919183271416048984895, −3.45772757471768282329224471522, −2.90280224646757478028790822878, −2.14618022176225219874554023158, −1.29481688602011781019288654733,
1.29481688602011781019288654733, 2.14618022176225219874554023158, 2.90280224646757478028790822878, 3.45772757471768282329224471522, 4.33367347919183271416048984895, 4.98605794679932717429857378222, 5.46808698406048069448159372901, 6.10334097807826589889646597817, 6.77965047928472316027355956812, 7.70203466342344005251643661101
