| L(s) = 1 | − 1.20·2-s − 1.29·3-s − 0.551·4-s + 5-s + 1.55·6-s − 0.246·7-s + 3.07·8-s − 1.32·9-s − 1.20·10-s − 1.77·11-s + 0.714·12-s − 3.33·13-s + 0.297·14-s − 1.29·15-s − 2.59·16-s + 1.21·17-s + 1.59·18-s + 1.20·19-s − 0.551·20-s + 0.319·21-s + 2.13·22-s − 3.82·23-s − 3.97·24-s + 25-s + 4.01·26-s + 5.59·27-s + 0.136·28-s + ⋯ |
| L(s) = 1 | − 0.850·2-s − 0.747·3-s − 0.275·4-s + 0.447·5-s + 0.636·6-s − 0.0933·7-s + 1.08·8-s − 0.440·9-s − 0.380·10-s − 0.534·11-s + 0.206·12-s − 0.925·13-s + 0.0794·14-s − 0.334·15-s − 0.647·16-s + 0.294·17-s + 0.375·18-s + 0.276·19-s − 0.123·20-s + 0.0697·21-s + 0.454·22-s − 0.798·23-s − 0.811·24-s + 0.200·25-s + 0.787·26-s + 1.07·27-s + 0.0257·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4706930099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4706930099\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 1.20T + 2T^{2} \) |
| 3 | \( 1 + 1.29T + 3T^{2} \) |
| 7 | \( 1 + 0.246T + 7T^{2} \) |
| 11 | \( 1 + 1.77T + 11T^{2} \) |
| 13 | \( 1 + 3.33T + 13T^{2} \) |
| 17 | \( 1 - 1.21T + 17T^{2} \) |
| 19 | \( 1 - 1.20T + 19T^{2} \) |
| 23 | \( 1 + 3.82T + 23T^{2} \) |
| 29 | \( 1 + 0.468T + 29T^{2} \) |
| 31 | \( 1 - 2.07T + 31T^{2} \) |
| 37 | \( 1 - 3.22T + 37T^{2} \) |
| 41 | \( 1 - 2.59T + 41T^{2} \) |
| 43 | \( 1 + 5.47T + 43T^{2} \) |
| 47 | \( 1 + 7.24T + 47T^{2} \) |
| 53 | \( 1 - 0.424T + 53T^{2} \) |
| 59 | \( 1 - 1.57T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 5.49T + 71T^{2} \) |
| 73 | \( 1 - 2.36T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 - 2.50T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
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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
−9.357808248775573963729628541048, −8.318653377968691126250842674731, −7.82964499982824253741011038705, −6.85938934622849710881203954160, −5.96218263754913824135438121486, −5.16483367347119436720148398246, −4.55757720583727190235327914780, −3.13012864822897396052181120752, −1.92510765655015339981117822958, −0.53023084903984358147785668654,
0.53023084903984358147785668654, 1.92510765655015339981117822958, 3.13012864822897396052181120752, 4.55757720583727190235327914780, 5.16483367347119436720148398246, 5.96218263754913824135438121486, 6.85938934622849710881203954160, 7.82964499982824253741011038705, 8.318653377968691126250842674731, 9.357808248775573963729628541048
