| L(s) = 1 | − 2-s + 4-s − 5-s + 3.09·7-s − 8-s + 10-s + 11-s − 2.45·13-s − 3.09·14-s + 16-s + 6.46·17-s + 3.73·19-s − 20-s − 22-s − 6.12·23-s + 25-s + 2.45·26-s + 3.09·28-s − 4.09·29-s − 0.00993·31-s − 32-s − 6.46·34-s − 3.09·35-s − 0.846·37-s − 3.73·38-s + 40-s − 5.69·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.17·7-s − 0.353·8-s + 0.316·10-s + 0.301·11-s − 0.680·13-s − 0.827·14-s + 0.250·16-s + 1.56·17-s + 0.856·19-s − 0.223·20-s − 0.213·22-s − 1.27·23-s + 0.200·25-s + 0.481·26-s + 0.585·28-s − 0.760·29-s − 0.00178·31-s − 0.176·32-s − 1.10·34-s − 0.523·35-s − 0.139·37-s − 0.605·38-s + 0.158·40-s − 0.889·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8910 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8910 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 3.09T + 7T^{2} \) |
| 13 | \( 1 + 2.45T + 13T^{2} \) |
| 17 | \( 1 - 6.46T + 17T^{2} \) |
| 19 | \( 1 - 3.73T + 19T^{2} \) |
| 23 | \( 1 + 6.12T + 23T^{2} \) |
| 29 | \( 1 + 4.09T + 29T^{2} \) |
| 31 | \( 1 + 0.00993T + 31T^{2} \) |
| 37 | \( 1 + 0.846T + 37T^{2} \) |
| 41 | \( 1 + 5.69T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 + 0.781T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 3.60T + 59T^{2} \) |
| 61 | \( 1 - 3.78T + 61T^{2} \) |
| 67 | \( 1 - 4.66T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 + 7.94T + 73T^{2} \) |
| 79 | \( 1 - 7.21T + 79T^{2} \) |
| 83 | \( 1 + 2.23T + 83T^{2} \) |
| 89 | \( 1 + 8.33T + 89T^{2} \) |
| 97 | \( 1 - 9.11T + 97T^{2} \) |
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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
−7.65012024477468867549269592670, −7.00583996343374536123453778084, −6.06052326219543042603472534863, −5.29899801174485232283244286489, −4.73945844660769079851895105676, −3.71663088331128769496260909929, −3.06061911489754762851407604813, −1.86763974838319069068878613409, −1.30832889026459777698731407741, 0,
1.30832889026459777698731407741, 1.86763974838319069068878613409, 3.06061911489754762851407604813, 3.71663088331128769496260909929, 4.73945844660769079851895105676, 5.29899801174485232283244286489, 6.06052326219543042603472534863, 7.00583996343374536123453778084, 7.65012024477468867549269592670
