| L(s) = 1 | + 2.73·2-s + 5.50·4-s + 2.48·5-s + 2.47·7-s + 9.58·8-s + 6.81·10-s − 3.17·11-s + 6.77·13-s + 6.79·14-s + 15.2·16-s − 3.00·17-s − 6.29·19-s + 13.6·20-s − 8.68·22-s − 2.42·23-s + 1.19·25-s + 18.5·26-s + 13.6·28-s + 0.314·29-s − 7.36·31-s + 22.5·32-s − 8.24·34-s + 6.17·35-s − 10.4·37-s − 17.2·38-s + 23.8·40-s + 6.72·41-s + ⋯ |
| L(s) = 1 | + 1.93·2-s + 2.75·4-s + 1.11·5-s + 0.937·7-s + 3.38·8-s + 2.15·10-s − 0.955·11-s + 1.88·13-s + 1.81·14-s + 3.81·16-s − 0.729·17-s − 1.44·19-s + 3.06·20-s − 1.85·22-s − 0.505·23-s + 0.238·25-s + 3.64·26-s + 2.57·28-s + 0.0584·29-s − 1.32·31-s + 3.99·32-s − 1.41·34-s + 1.04·35-s − 1.72·37-s − 2.79·38-s + 3.77·40-s + 1.05·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.295567892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.295567892\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 2 | \( 1 - 2.73T + 2T^{2} \) |
| 5 | \( 1 - 2.48T + 5T^{2} \) |
| 7 | \( 1 - 2.47T + 7T^{2} \) |
| 11 | \( 1 + 3.17T + 11T^{2} \) |
| 13 | \( 1 - 6.77T + 13T^{2} \) |
| 17 | \( 1 + 3.00T + 17T^{2} \) |
| 19 | \( 1 + 6.29T + 19T^{2} \) |
| 23 | \( 1 + 2.42T + 23T^{2} \) |
| 29 | \( 1 - 0.314T + 29T^{2} \) |
| 31 | \( 1 + 7.36T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 - 6.72T + 41T^{2} \) |
| 43 | \( 1 + 7.53T + 43T^{2} \) |
| 47 | \( 1 - 8.67T + 47T^{2} \) |
| 53 | \( 1 + 4.75T + 53T^{2} \) |
| 59 | \( 1 + 6.54T + 59T^{2} \) |
| 61 | \( 1 + 8.08T + 61T^{2} \) |
| 67 | \( 1 + 7.75T + 67T^{2} \) |
| 71 | \( 1 + 5.97T + 71T^{2} \) |
| 73 | \( 1 - 1.31T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 - 4.52T + 83T^{2} \) |
| 89 | \( 1 + 4.07T + 89T^{2} \) |
| 97 | \( 1 - 8.45T + 97T^{2} \) |
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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
−8.268761786472249436158486613495, −7.46022881909405303478377692441, −6.47452323863992879653060885349, −6.03474158848026546361445853943, −5.45899956396345644320316004471, −4.71686869448008605954559056758, −4.01117382665686671732372190221, −3.13539905916066264922006009150, −1.93784807911743257685618811951, −1.80476900094142115354811006166,
1.80476900094142115354811006166, 1.93784807911743257685618811951, 3.13539905916066264922006009150, 4.01117382665686671732372190221, 4.71686869448008605954559056758, 5.45899956396345644320316004471, 6.03474158848026546361445853943, 6.47452323863992879653060885349, 7.46022881909405303478377692441, 8.268761786472249436158486613495
