| L(s) = 1 | + 2-s + 2.71·3-s + 4-s − 1.49·5-s + 2.71·6-s − 4.64·7-s + 8-s + 4.38·9-s − 1.49·10-s − 0.321·11-s + 2.71·12-s + 4.03·13-s − 4.64·14-s − 4.06·15-s + 16-s − 6.72·17-s + 4.38·18-s − 0.992·19-s − 1.49·20-s − 12.6·21-s − 0.321·22-s − 23-s + 2.71·24-s − 2.76·25-s + 4.03·26-s + 3.75·27-s − 4.64·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.56·3-s + 0.5·4-s − 0.668·5-s + 1.10·6-s − 1.75·7-s + 0.353·8-s + 1.46·9-s − 0.472·10-s − 0.0970·11-s + 0.784·12-s + 1.11·13-s − 1.24·14-s − 1.04·15-s + 0.250·16-s − 1.63·17-s + 1.03·18-s − 0.227·19-s − 0.334·20-s − 2.75·21-s − 0.0686·22-s − 0.208·23-s + 0.554·24-s − 0.552·25-s + 0.790·26-s + 0.723·27-s − 0.878·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 2.71T + 3T^{2} \) |
| 5 | \( 1 + 1.49T + 5T^{2} \) |
| 7 | \( 1 + 4.64T + 7T^{2} \) |
| 11 | \( 1 + 0.321T + 11T^{2} \) |
| 13 | \( 1 - 4.03T + 13T^{2} \) |
| 17 | \( 1 + 6.72T + 17T^{2} \) |
| 19 | \( 1 + 0.992T + 19T^{2} \) |
| 29 | \( 1 + 8.49T + 29T^{2} \) |
| 31 | \( 1 - 2.09T + 31T^{2} \) |
| 37 | \( 1 - 1.74T + 37T^{2} \) |
| 41 | \( 1 + 0.703T + 41T^{2} \) |
| 43 | \( 1 - 5.32T + 43T^{2} \) |
| 47 | \( 1 + 3.48T + 47T^{2} \) |
| 53 | \( 1 - 1.68T + 53T^{2} \) |
| 59 | \( 1 - 1.19T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 - 3.87T + 71T^{2} \) |
| 73 | \( 1 + 2.17T + 73T^{2} \) |
| 79 | \( 1 + 5.60T + 79T^{2} \) |
| 83 | \( 1 + 3.90T + 83T^{2} \) |
| 89 | \( 1 + 1.79T + 89T^{2} \) |
| 97 | \( 1 + 2.15T + 97T^{2} \) |
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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
−7.70223994009308459203599028089, −7.03213061732725675402041726267, −6.37718675427451569077665447357, −5.73920115665213484132502404607, −4.19804630005784473357262158904, −4.02874101641391069394075511223, −3.23962694413159573257102173038, −2.71172122377492033542761586689, −1.77953495965601595336043604897, 0,
1.77953495965601595336043604897, 2.71172122377492033542761586689, 3.23962694413159573257102173038, 4.02874101641391069394075511223, 4.19804630005784473357262158904, 5.73920115665213484132502404607, 6.37718675427451569077665447357, 7.03213061732725675402041726267, 7.70223994009308459203599028089
