| L(s) = 1 | − 0.801·2-s − 1.69·3-s − 1.35·4-s + 1.35·6-s + 7-s + 2.69·8-s − 0.137·9-s − 3.80·11-s + 2.29·12-s + 4.54·13-s − 0.801·14-s + 0.554·16-s − 2.91·17-s + 0.109·18-s − 3.35·19-s − 1.69·21-s + 3.04·22-s + 23-s − 4.55·24-s − 3.64·26-s + 5.30·27-s − 1.35·28-s + 2.09·29-s + 0.246·31-s − 5.82·32-s + 6.43·33-s + 2.33·34-s + ⋯ |
| L(s) = 1 | − 0.567·2-s − 0.976·3-s − 0.678·4-s + 0.553·6-s + 0.377·7-s + 0.951·8-s − 0.0456·9-s − 1.14·11-s + 0.662·12-s + 1.25·13-s − 0.214·14-s + 0.138·16-s − 0.706·17-s + 0.0259·18-s − 0.770·19-s − 0.369·21-s + 0.650·22-s + 0.208·23-s − 0.929·24-s − 0.714·26-s + 1.02·27-s − 0.256·28-s + 0.389·29-s + 0.0443·31-s − 1.03·32-s + 1.11·33-s + 0.400·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 0.801T + 2T^{2} \) |
| 3 | \( 1 + 1.69T + 3T^{2} \) |
| 11 | \( 1 + 3.80T + 11T^{2} \) |
| 13 | \( 1 - 4.54T + 13T^{2} \) |
| 17 | \( 1 + 2.91T + 17T^{2} \) |
| 19 | \( 1 + 3.35T + 19T^{2} \) |
| 29 | \( 1 - 2.09T + 29T^{2} \) |
| 31 | \( 1 - 0.246T + 31T^{2} \) |
| 37 | \( 1 + 5.40T + 37T^{2} \) |
| 41 | \( 1 - 2.82T + 41T^{2} \) |
| 43 | \( 1 - 0.801T + 43T^{2} \) |
| 47 | \( 1 + 1.97T + 47T^{2} \) |
| 53 | \( 1 - 9.78T + 53T^{2} \) |
| 59 | \( 1 + 6.66T + 59T^{2} \) |
| 61 | \( 1 + 5.02T + 61T^{2} \) |
| 67 | \( 1 - 5.03T + 67T^{2} \) |
| 71 | \( 1 - 5.05T + 71T^{2} \) |
| 73 | \( 1 - 0.972T + 73T^{2} \) |
| 79 | \( 1 - 3.14T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 - 7.18T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−8.377314658512906070524025900943, −7.46006098675106934615307436057, −6.53052063550509640138981441180, −5.81300856214236181626568139921, −5.07975734053058104781991530741, −4.52288105134611577803676430120, −3.53650762062777632586206897457, −2.23753790580279990773571128292, −1.00534766521534650807430766966, 0,
1.00534766521534650807430766966, 2.23753790580279990773571128292, 3.53650762062777632586206897457, 4.52288105134611577803676430120, 5.07975734053058104781991530741, 5.81300856214236181626568139921, 6.53052063550509640138981441180, 7.46006098675106934615307436057, 8.377314658512906070524025900943
