| L(s) = 1 | − 2.32·2-s + 1.35·3-s + 3.38·4-s + 2.79·5-s − 3.14·6-s + 4.55·7-s − 3.21·8-s − 1.16·9-s − 6.48·10-s − 4.50·11-s + 4.58·12-s − 0.542·13-s − 10.5·14-s + 3.78·15-s + 0.681·16-s − 5.71·17-s + 2.70·18-s − 1.76·19-s + 9.45·20-s + 6.17·21-s + 10.4·22-s − 3.18·23-s − 4.34·24-s + 2.81·25-s + 1.25·26-s − 5.64·27-s + 15.4·28-s + ⋯ |
| L(s) = 1 | − 1.64·2-s + 0.781·3-s + 1.69·4-s + 1.24·5-s − 1.28·6-s + 1.72·7-s − 1.13·8-s − 0.388·9-s − 2.05·10-s − 1.35·11-s + 1.32·12-s − 0.150·13-s − 2.82·14-s + 0.977·15-s + 0.170·16-s − 1.38·17-s + 0.637·18-s − 0.404·19-s + 2.11·20-s + 1.34·21-s + 2.22·22-s − 0.663·23-s − 0.887·24-s + 0.562·25-s + 0.247·26-s − 1.08·27-s + 2.91·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 37 | \( 1 + T \) |
| 109 | \( 1 + T \) |
| good | 2 | \( 1 + 2.32T + 2T^{2} \) |
| 3 | \( 1 - 1.35T + 3T^{2} \) |
| 5 | \( 1 - 2.79T + 5T^{2} \) |
| 7 | \( 1 - 4.55T + 7T^{2} \) |
| 11 | \( 1 + 4.50T + 11T^{2} \) |
| 13 | \( 1 + 0.542T + 13T^{2} \) |
| 17 | \( 1 + 5.71T + 17T^{2} \) |
| 19 | \( 1 + 1.76T + 19T^{2} \) |
| 23 | \( 1 + 3.18T + 23T^{2} \) |
| 29 | \( 1 + 7.61T + 29T^{2} \) |
| 31 | \( 1 + 1.83T + 31T^{2} \) |
| 41 | \( 1 + 9.61T + 41T^{2} \) |
| 43 | \( 1 - 3.29T + 43T^{2} \) |
| 47 | \( 1 - 7.88T + 47T^{2} \) |
| 53 | \( 1 + 3.17T + 53T^{2} \) |
| 59 | \( 1 - 1.31T + 59T^{2} \) |
| 61 | \( 1 - 2.66T + 61T^{2} \) |
| 67 | \( 1 + 3.57T + 67T^{2} \) |
| 71 | \( 1 + 8.88T + 71T^{2} \) |
| 73 | \( 1 - 4.02T + 73T^{2} \) |
| 79 | \( 1 + 17.5T + 79T^{2} \) |
| 83 | \( 1 - 8.15T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 0.623T + 97T^{2} \) |
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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
−8.328680763239147820661401649154, −7.67949062150438840658958017743, −7.07027397166315305440511462497, −5.89425425668066058046661890804, −5.29962703875739745990154084850, −4.30706482090464852502791915860, −2.69174173559284022060036200886, −2.04004212948496494104133229046, −1.74265433208672746095731924528, 0,
1.74265433208672746095731924528, 2.04004212948496494104133229046, 2.69174173559284022060036200886, 4.30706482090464852502791915860, 5.29962703875739745990154084850, 5.89425425668066058046661890804, 7.07027397166315305440511462497, 7.67949062150438840658958017743, 8.328680763239147820661401649154
