| L(s) = 1 | + 2-s + 1.74·3-s + 4-s − 1.04·5-s + 1.74·6-s − 2.93·7-s + 8-s + 0.0467·9-s − 1.04·10-s − 0.729·11-s + 1.74·12-s + 1.00·13-s − 2.93·14-s − 1.82·15-s + 16-s − 1.00·17-s + 0.0467·18-s + 0.315·19-s − 1.04·20-s − 5.12·21-s − 0.729·22-s + 0.448·23-s + 1.74·24-s − 3.90·25-s + 1.00·26-s − 5.15·27-s − 2.93·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.00·3-s + 0.5·4-s − 0.468·5-s + 0.712·6-s − 1.11·7-s + 0.353·8-s + 0.0155·9-s − 0.331·10-s − 0.219·11-s + 0.503·12-s + 0.279·13-s − 0.785·14-s − 0.471·15-s + 0.250·16-s − 0.244·17-s + 0.0110·18-s + 0.0724·19-s − 0.234·20-s − 1.11·21-s − 0.155·22-s + 0.0934·23-s + 0.356·24-s − 0.780·25-s + 0.197·26-s − 0.992·27-s − 0.555·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 - 1.74T + 3T^{2} \) |
| 5 | \( 1 + 1.04T + 5T^{2} \) |
| 7 | \( 1 + 2.93T + 7T^{2} \) |
| 11 | \( 1 + 0.729T + 11T^{2} \) |
| 13 | \( 1 - 1.00T + 13T^{2} \) |
| 17 | \( 1 + 1.00T + 17T^{2} \) |
| 19 | \( 1 - 0.315T + 19T^{2} \) |
| 23 | \( 1 - 0.448T + 23T^{2} \) |
| 29 | \( 1 + 6.89T + 29T^{2} \) |
| 31 | \( 1 + 1.10T + 31T^{2} \) |
| 37 | \( 1 - 1.58T + 37T^{2} \) |
| 41 | \( 1 - 6.58T + 41T^{2} \) |
| 43 | \( 1 + 4.46T + 43T^{2} \) |
| 47 | \( 1 + 8.65T + 47T^{2} \) |
| 53 | \( 1 + 9.09T + 53T^{2} \) |
| 59 | \( 1 + 0.460T + 59T^{2} \) |
| 61 | \( 1 + 3.78T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 - 8.18T + 71T^{2} \) |
| 73 | \( 1 + 0.125T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 + 1.99T + 83T^{2} \) |
| 89 | \( 1 + 5.47T + 89T^{2} \) |
| 97 | \( 1 - 5.48T + 97T^{2} \) |
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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
−7.86199686860942284687331058441, −7.53594551248887493883381647248, −6.48281726114898462982809083397, −5.95899669487737223403121208374, −4.97641294787617170126238367662, −3.92994018896876874143156264553, −3.44323402994159273651357221279, −2.78780777617799674293225595921, −1.82772320259222513870878595976, 0,
1.82772320259222513870878595976, 2.78780777617799674293225595921, 3.44323402994159273651357221279, 3.92994018896876874143156264553, 4.97641294787617170126238367662, 5.95899669487737223403121208374, 6.48281726114898462982809083397, 7.53594551248887493883381647248, 7.86199686860942284687331058441
