| L(s) = 1 | + 0.489·2-s − 1.75·4-s + 2.01·5-s + 0.633·7-s − 1.84·8-s + 0.986·10-s − 6.54·11-s + 3.56·13-s + 0.310·14-s + 2.61·16-s + 7.01·17-s − 19-s − 3.54·20-s − 3.20·22-s − 6.64·23-s − 0.948·25-s + 1.74·26-s − 1.11·28-s + 7.37·29-s − 4.60·31-s + 4.96·32-s + 3.43·34-s + 1.27·35-s − 1.41·37-s − 0.489·38-s − 3.70·40-s − 4.14·41-s + ⋯ |
| L(s) = 1 | + 0.346·2-s − 0.879·4-s + 0.900·5-s + 0.239·7-s − 0.651·8-s + 0.311·10-s − 1.97·11-s + 0.987·13-s + 0.0829·14-s + 0.654·16-s + 1.70·17-s − 0.229·19-s − 0.792·20-s − 0.684·22-s − 1.38·23-s − 0.189·25-s + 0.342·26-s − 0.210·28-s + 1.36·29-s − 0.827·31-s + 0.878·32-s + 0.589·34-s + 0.215·35-s − 0.231·37-s − 0.0794·38-s − 0.586·40-s − 0.647·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 0.489T + 2T^{2} \) |
| 5 | \( 1 - 2.01T + 5T^{2} \) |
| 7 | \( 1 - 0.633T + 7T^{2} \) |
| 11 | \( 1 + 6.54T + 11T^{2} \) |
| 13 | \( 1 - 3.56T + 13T^{2} \) |
| 17 | \( 1 - 7.01T + 17T^{2} \) |
| 23 | \( 1 + 6.64T + 23T^{2} \) |
| 29 | \( 1 - 7.37T + 29T^{2} \) |
| 31 | \( 1 + 4.60T + 31T^{2} \) |
| 37 | \( 1 + 1.41T + 37T^{2} \) |
| 41 | \( 1 + 4.14T + 41T^{2} \) |
| 43 | \( 1 + 8.19T + 43T^{2} \) |
| 53 | \( 1 - 3.29T + 53T^{2} \) |
| 59 | \( 1 + 1.03T + 59T^{2} \) |
| 61 | \( 1 - 7.97T + 61T^{2} \) |
| 67 | \( 1 - 9.33T + 67T^{2} \) |
| 71 | \( 1 - 7.70T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 - 5.64T + 79T^{2} \) |
| 83 | \( 1 - 2.87T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−7.77674904080201379700536539685, −6.58170513864808351734857383119, −5.77840555776541184331584020165, −5.42368016034552414866181969922, −4.90571408020765673168212706917, −3.85862556643874282740038822667, −3.20569629035813499911688720281, −2.30264004906418539143308663017, −1.28365383489578617910973312167, 0,
1.28365383489578617910973312167, 2.30264004906418539143308663017, 3.20569629035813499911688720281, 3.85862556643874282740038822667, 4.90571408020765673168212706917, 5.42368016034552414866181969922, 5.77840555776541184331584020165, 6.58170513864808351734857383119, 7.77674904080201379700536539685
