| L(s) = 1 | + 1.19·2-s − 0.566·4-s + 2.06·5-s − 5.04·7-s − 3.07·8-s + 2.46·10-s + 3.25·11-s − 6.23·13-s − 6.03·14-s − 2.54·16-s + 1.68·17-s − 6.81·19-s − 1.16·20-s + 3.89·22-s − 23-s − 0.747·25-s − 7.47·26-s + 2.85·28-s − 29-s + 7.68·31-s + 3.09·32-s + 2.02·34-s − 10.3·35-s + 8.10·37-s − 8.15·38-s − 6.33·40-s + 3.55·41-s + ⋯ |
| L(s) = 1 | + 0.846·2-s − 0.283·4-s + 0.922·5-s − 1.90·7-s − 1.08·8-s + 0.780·10-s + 0.980·11-s − 1.73·13-s − 1.61·14-s − 0.636·16-s + 0.409·17-s − 1.56·19-s − 0.261·20-s + 0.829·22-s − 0.208·23-s − 0.149·25-s − 1.46·26-s + 0.539·28-s − 0.185·29-s + 1.38·31-s + 0.547·32-s + 0.346·34-s − 1.75·35-s + 1.33·37-s − 1.32·38-s − 1.00·40-s + 0.555·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.656229194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.656229194\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.19T + 2T^{2} \) |
| 5 | \( 1 - 2.06T + 5T^{2} \) |
| 7 | \( 1 + 5.04T + 7T^{2} \) |
| 11 | \( 1 - 3.25T + 11T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 - 1.68T + 17T^{2} \) |
| 19 | \( 1 + 6.81T + 19T^{2} \) |
| 31 | \( 1 - 7.68T + 31T^{2} \) |
| 37 | \( 1 - 8.10T + 37T^{2} \) |
| 41 | \( 1 - 3.55T + 41T^{2} \) |
| 43 | \( 1 + 5.36T + 43T^{2} \) |
| 47 | \( 1 - 2.48T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 - 6.25T + 59T^{2} \) |
| 61 | \( 1 + 5.81T + 61T^{2} \) |
| 67 | \( 1 - 8.79T + 67T^{2} \) |
| 71 | \( 1 + 3.86T + 71T^{2} \) |
| 73 | \( 1 - 4.08T + 73T^{2} \) |
| 79 | \( 1 - 3.99T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 - 6.20T + 89T^{2} \) |
| 97 | \( 1 + 7.98T + 97T^{2} \) |
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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
−8.110306635774061575865886756241, −6.93820471192534811093002525757, −6.45930680471344848374756499902, −5.99626216250044623293448475546, −5.27956117047105338387530621630, −4.30889902641244445765322007765, −3.81604278953805532641583325986, −2.77513185461596995122791028566, −2.32331294405965750904522473815, −0.56148435626708355565092382162,
0.56148435626708355565092382162, 2.32331294405965750904522473815, 2.77513185461596995122791028566, 3.81604278953805532641583325986, 4.30889902641244445765322007765, 5.27956117047105338387530621630, 5.99626216250044623293448475546, 6.45930680471344848374756499902, 6.93820471192534811093002525757, 8.110306635774061575865886756241
