| L(s) = 1 | + 2-s − 1.55·3-s + 4-s + 5-s − 1.55·6-s + 2.74·7-s + 8-s − 0.579·9-s + 10-s + 2.55·11-s − 1.55·12-s + 2.55·13-s + 2.74·14-s − 1.55·15-s + 16-s + 1.70·17-s − 0.579·18-s − 5.75·19-s + 20-s − 4.27·21-s + 2.55·22-s − 2.92·23-s − 1.55·24-s + 25-s + 2.55·26-s + 5.56·27-s + 2.74·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.898·3-s + 0.5·4-s + 0.447·5-s − 0.635·6-s + 1.03·7-s + 0.353·8-s − 0.193·9-s + 0.316·10-s + 0.771·11-s − 0.449·12-s + 0.707·13-s + 0.733·14-s − 0.401·15-s + 0.250·16-s + 0.413·17-s − 0.136·18-s − 1.32·19-s + 0.223·20-s − 0.932·21-s + 0.545·22-s − 0.609·23-s − 0.317·24-s + 0.200·25-s + 0.500·26-s + 1.07·27-s + 0.518·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.074909279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.074909279\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 601 | \( 1 - T \) |
| good | 3 | \( 1 + 1.55T + 3T^{2} \) |
| 7 | \( 1 - 2.74T + 7T^{2} \) |
| 11 | \( 1 - 2.55T + 11T^{2} \) |
| 13 | \( 1 - 2.55T + 13T^{2} \) |
| 17 | \( 1 - 1.70T + 17T^{2} \) |
| 19 | \( 1 + 5.75T + 19T^{2} \) |
| 23 | \( 1 + 2.92T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 - 2.19T + 31T^{2} \) |
| 37 | \( 1 + 4.12T + 37T^{2} \) |
| 41 | \( 1 - 2.67T + 41T^{2} \) |
| 43 | \( 1 - 8.80T + 43T^{2} \) |
| 47 | \( 1 - 2.28T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 - 1.28T + 59T^{2} \) |
| 61 | \( 1 - 9.98T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 8.10T + 71T^{2} \) |
| 73 | \( 1 - 5.40T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 4.17T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−8.238064795092467930119704112030, −7.00393811707409649244676403417, −6.51961984544346525443828139691, −5.70067279105675627180162238936, −5.46649261618866324978155085679, −4.40030464161080386739254224501, −4.01919000843684920081197034228, −2.75501560419306651448425047714, −1.83604171788483790605350053762, −0.911959105637846248518413543578,
0.911959105637846248518413543578, 1.83604171788483790605350053762, 2.75501560419306651448425047714, 4.01919000843684920081197034228, 4.40030464161080386739254224501, 5.46649261618866324978155085679, 5.70067279105675627180162238936, 6.51961984544346525443828139691, 7.00393811707409649244676403417, 8.238064795092467930119704112030
