| L(s) = 1 | − 1.76·2-s + 1.12·4-s + 2.50·5-s − 4.08·7-s + 1.55·8-s − 4.42·10-s + 11-s + 4.58·13-s + 7.22·14-s − 4.98·16-s + 1.68·17-s + 2.54·19-s + 2.80·20-s − 1.76·22-s − 6.64·23-s + 1.26·25-s − 8.09·26-s − 4.58·28-s − 2.87·29-s + 5.77·31-s + 5.69·32-s − 2.98·34-s − 10.2·35-s + 10.5·37-s − 4.49·38-s + 3.89·40-s + 7.50·41-s + ⋯ |
| L(s) = 1 | − 1.24·2-s + 0.560·4-s + 1.11·5-s − 1.54·7-s + 0.549·8-s − 1.39·10-s + 0.301·11-s + 1.27·13-s + 1.93·14-s − 1.24·16-s + 0.409·17-s + 0.583·19-s + 0.626·20-s − 0.376·22-s − 1.38·23-s + 0.252·25-s − 1.58·26-s − 0.865·28-s − 0.533·29-s + 1.03·31-s + 1.00·32-s − 0.511·34-s − 1.72·35-s + 1.73·37-s − 0.728·38-s + 0.615·40-s + 1.17·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.035164585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.035164585\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 1.76T + 2T^{2} \) |
| 5 | \( 1 - 2.50T + 5T^{2} \) |
| 7 | \( 1 + 4.08T + 7T^{2} \) |
| 13 | \( 1 - 4.58T + 13T^{2} \) |
| 17 | \( 1 - 1.68T + 17T^{2} \) |
| 19 | \( 1 - 2.54T + 19T^{2} \) |
| 23 | \( 1 + 6.64T + 23T^{2} \) |
| 29 | \( 1 + 2.87T + 29T^{2} \) |
| 31 | \( 1 - 5.77T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 7.50T + 41T^{2} \) |
| 43 | \( 1 + 5.62T + 43T^{2} \) |
| 47 | \( 1 + 8.24T + 47T^{2} \) |
| 53 | \( 1 - 6.73T + 53T^{2} \) |
| 59 | \( 1 + 1.44T + 59T^{2} \) |
| 67 | \( 1 + 2.89T + 67T^{2} \) |
| 71 | \( 1 - 3.64T + 71T^{2} \) |
| 73 | \( 1 + 1.53T + 73T^{2} \) |
| 79 | \( 1 - 6.59T + 79T^{2} \) |
| 83 | \( 1 + 4.68T + 83T^{2} \) |
| 89 | \( 1 + 8.01T + 89T^{2} \) |
| 97 | \( 1 + 8.48T + 97T^{2} \) |
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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
−8.195918101616088783421692740494, −7.54870807292454596588395071308, −6.53532912928901095546535540561, −6.21042397070028902431211913954, −5.58570922285244273010918236277, −4.30720340134000214990759733727, −3.49922457885070717152230188482, −2.54003045720345250237665352345, −1.57545068657445943036227458046, −0.67828378037022359312773144964,
0.67828378037022359312773144964, 1.57545068657445943036227458046, 2.54003045720345250237665352345, 3.49922457885070717152230188482, 4.30720340134000214990759733727, 5.58570922285244273010918236277, 6.21042397070028902431211913954, 6.53532912928901095546535540561, 7.54870807292454596588395071308, 8.195918101616088783421692740494
