| L(s) = 1 | − 2-s − 2.86·3-s + 4-s − 5-s + 2.86·6-s + 0.981·7-s − 8-s + 5.21·9-s + 10-s + 3.23·11-s − 2.86·12-s − 5.40·13-s − 0.981·14-s + 2.86·15-s + 16-s − 3.01·17-s − 5.21·18-s + 4.93·19-s − 20-s − 2.81·21-s − 3.23·22-s + 1.30·23-s + 2.86·24-s + 25-s + 5.40·26-s − 6.34·27-s + 0.981·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.65·3-s + 0.5·4-s − 0.447·5-s + 1.17·6-s + 0.370·7-s − 0.353·8-s + 1.73·9-s + 0.316·10-s + 0.975·11-s − 0.827·12-s − 1.49·13-s − 0.262·14-s + 0.740·15-s + 0.250·16-s − 0.731·17-s − 1.22·18-s + 1.13·19-s − 0.223·20-s − 0.613·21-s − 0.690·22-s + 0.271·23-s + 0.585·24-s + 0.200·25-s + 1.05·26-s − 1.22·27-s + 0.185·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 + 2.86T + 3T^{2} \) |
| 7 | \( 1 - 0.981T + 7T^{2} \) |
| 11 | \( 1 - 3.23T + 11T^{2} \) |
| 13 | \( 1 + 5.40T + 13T^{2} \) |
| 17 | \( 1 + 3.01T + 17T^{2} \) |
| 19 | \( 1 - 4.93T + 19T^{2} \) |
| 23 | \( 1 - 1.30T + 23T^{2} \) |
| 29 | \( 1 + 4.83T + 29T^{2} \) |
| 31 | \( 1 + 1.22T + 31T^{2} \) |
| 37 | \( 1 + 1.55T + 37T^{2} \) |
| 41 | \( 1 - 3.26T + 41T^{2} \) |
| 43 | \( 1 + 8.01T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 3.38T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 - 15.0T + 61T^{2} \) |
| 67 | \( 1 + 4.93T + 67T^{2} \) |
| 71 | \( 1 + 0.116T + 71T^{2} \) |
| 73 | \( 1 - 4.74T + 73T^{2} \) |
| 79 | \( 1 + 0.914T + 79T^{2} \) |
| 83 | \( 1 - 7.48T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−7.84566114631201486642038122048, −7.24906478306315004966517926720, −6.74872388659980368164276902774, −5.93861543616459008116505094135, −5.09625153256857087509830483257, −4.58675728659990194922583953676, −3.49814494130326947372406443804, −2.09177702485041161397186369010, −1.01811156127688130843557216962, 0,
1.01811156127688130843557216962, 2.09177702485041161397186369010, 3.49814494130326947372406443804, 4.58675728659990194922583953676, 5.09625153256857087509830483257, 5.93861543616459008116505094135, 6.74872388659980368164276902774, 7.24906478306315004966517926720, 7.84566114631201486642038122048
