| L(s) = 1 | + 0.865·2-s − 1.25·4-s − 1.66·5-s − 0.715·7-s − 2.81·8-s − 1.44·10-s + 0.656·11-s − 4.37·13-s − 0.619·14-s + 0.0635·16-s + 7.15·17-s + 6.56·19-s + 2.08·20-s + 0.568·22-s + 23-s − 2.22·25-s − 3.78·26-s + 0.895·28-s − 29-s + 0.705·31-s + 5.68·32-s + 6.19·34-s + 1.19·35-s − 7.06·37-s + 5.68·38-s + 4.68·40-s + 12.6·41-s + ⋯ |
| L(s) = 1 | + 0.612·2-s − 0.625·4-s − 0.744·5-s − 0.270·7-s − 0.995·8-s − 0.456·10-s + 0.198·11-s − 1.21·13-s − 0.165·14-s + 0.0158·16-s + 1.73·17-s + 1.50·19-s + 0.465·20-s + 0.121·22-s + 0.208·23-s − 0.445·25-s − 0.742·26-s + 0.169·28-s − 0.185·29-s + 0.126·31-s + 1.00·32-s + 1.06·34-s + 0.201·35-s − 1.16·37-s + 0.922·38-s + 0.741·40-s + 1.97·41-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 0.865T + 2T^{2} \) |
| 5 | \( 1 + 1.66T + 5T^{2} \) |
| 7 | \( 1 + 0.715T + 7T^{2} \) |
| 11 | \( 1 - 0.656T + 11T^{2} \) |
| 13 | \( 1 + 4.37T + 13T^{2} \) |
| 17 | \( 1 - 7.15T + 17T^{2} \) |
| 19 | \( 1 - 6.56T + 19T^{2} \) |
| 31 | \( 1 - 0.705T + 31T^{2} \) |
| 37 | \( 1 + 7.06T + 37T^{2} \) |
| 41 | \( 1 - 12.6T + 41T^{2} \) |
| 43 | \( 1 + 2.05T + 43T^{2} \) |
| 47 | \( 1 + 8.50T + 47T^{2} \) |
| 53 | \( 1 + 3.04T + 53T^{2} \) |
| 59 | \( 1 - 4.81T + 59T^{2} \) |
| 61 | \( 1 - 2.51T + 61T^{2} \) |
| 67 | \( 1 + 2.81T + 67T^{2} \) |
| 71 | \( 1 + 0.253T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 0.611T + 79T^{2} \) |
| 83 | \( 1 + 2.06T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−7.76711808248047661322157499356, −7.11521561295381437324358030543, −6.14596535871984406553984804977, −5.26337532072733495744322631012, −5.00826004247202431638756783803, −3.92202123852970200808419592819, −3.43893247969788660313489150792, −2.71738633746387296269068585392, −1.15307476771746518040977329413, 0,
1.15307476771746518040977329413, 2.71738633746387296269068585392, 3.43893247969788660313489150792, 3.92202123852970200808419592819, 5.00826004247202431638756783803, 5.26337532072733495744322631012, 6.14596535871984406553984804977, 7.11521561295381437324358030543, 7.76711808248047661322157499356
