| L(s) = 1 | − 2.24·2-s + 3.05·4-s + 0.269·5-s − 0.523·7-s − 2.36·8-s − 0.605·10-s − 5.54·11-s + 3.85·13-s + 1.17·14-s − 0.785·16-s + 1.23·17-s + 1.15·19-s + 0.822·20-s + 12.4·22-s + 23-s − 4.92·25-s − 8.66·26-s − 1.59·28-s − 29-s + 1.02·31-s + 6.49·32-s − 2.77·34-s − 0.141·35-s − 3.22·37-s − 2.59·38-s − 0.637·40-s + 3.67·41-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.52·4-s + 0.120·5-s − 0.197·7-s − 0.836·8-s − 0.191·10-s − 1.67·11-s + 1.06·13-s + 0.314·14-s − 0.196·16-s + 0.298·17-s + 0.265·19-s + 0.183·20-s + 2.65·22-s + 0.208·23-s − 0.985·25-s − 1.69·26-s − 0.302·28-s − 0.185·29-s + 0.183·31-s + 1.14·32-s − 0.475·34-s − 0.0238·35-s − 0.529·37-s − 0.421·38-s − 0.100·40-s + 0.573·41-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 | \( 1 + 2.24T + 2T^{2} \) |
| 5 | \( 1 - 0.269T + 5T^{2} \) |
| 7 | \( 1 + 0.523T + 7T^{2} \) |
| 11 | \( 1 + 5.54T + 11T^{2} \) |
| 13 | \( 1 - 3.85T + 13T^{2} \) |
| 17 | \( 1 - 1.23T + 17T^{2} \) |
| 19 | \( 1 - 1.15T + 19T^{2} \) |
| 31 | \( 1 - 1.02T + 31T^{2} \) |
| 37 | \( 1 + 3.22T + 37T^{2} \) |
| 41 | \( 1 - 3.67T + 41T^{2} \) |
| 43 | \( 1 - 2.77T + 43T^{2} \) |
| 47 | \( 1 - 6.71T + 47T^{2} \) |
| 53 | \( 1 - 2.58T + 53T^{2} \) |
| 59 | \( 1 - 9.36T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 + 2.14T + 67T^{2} \) |
| 71 | \( 1 + 4.70T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 + 8.53T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 3.00T + 97T^{2} \) |
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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
−7.74884349520729546397084711634, −7.47561507606019012463421016212, −6.47881934445695036301267776534, −5.79275194065132095464563475544, −5.01945626272103483529460124976, −3.87496052620505586903327247913, −2.87044207821374903326988219958, −2.09625837622109623231047969072, −1.07216330661801442861309088917, 0,
1.07216330661801442861309088917, 2.09625837622109623231047969072, 2.87044207821374903326988219958, 3.87496052620505586903327247913, 5.01945626272103483529460124976, 5.79275194065132095464563475544, 6.47881934445695036301267776534, 7.47561507606019012463421016212, 7.74884349520729546397084711634
