| L(s) = 1 | − 2-s − 0.211·3-s + 4-s − 5-s + 0.211·6-s + 7-s − 8-s − 2.95·9-s + 10-s − 0.211·12-s + 0.384·13-s − 14-s + 0.211·15-s + 16-s + 3.35·17-s + 2.95·18-s + 4.82·19-s − 20-s − 0.211·21-s + 0.683·23-s + 0.211·24-s + 25-s − 0.384·26-s + 1.25·27-s + 28-s − 1.60·29-s − 0.211·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.121·3-s + 0.5·4-s − 0.447·5-s + 0.0861·6-s + 0.377·7-s − 0.353·8-s − 0.985·9-s + 0.316·10-s − 0.0609·12-s + 0.106·13-s − 0.267·14-s + 0.0545·15-s + 0.250·16-s + 0.812·17-s + 0.696·18-s + 1.10·19-s − 0.223·20-s − 0.0460·21-s + 0.142·23-s + 0.0430·24-s + 0.200·25-s − 0.0753·26-s + 0.241·27-s + 0.188·28-s − 0.298·29-s − 0.0385·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 0.211T + 3T^{2} \) |
| 13 | \( 1 - 0.384T + 13T^{2} \) |
| 17 | \( 1 - 3.35T + 17T^{2} \) |
| 19 | \( 1 - 4.82T + 19T^{2} \) |
| 23 | \( 1 - 0.683T + 23T^{2} \) |
| 29 | \( 1 + 1.60T + 29T^{2} \) |
| 31 | \( 1 + 1.25T + 31T^{2} \) |
| 37 | \( 1 - 0.856T + 37T^{2} \) |
| 41 | \( 1 + 9.62T + 41T^{2} \) |
| 43 | \( 1 + 9.55T + 43T^{2} \) |
| 47 | \( 1 + 7.55T + 47T^{2} \) |
| 53 | \( 1 + 13.5T + 53T^{2} \) |
| 59 | \( 1 - 7.26T + 59T^{2} \) |
| 61 | \( 1 + 3.89T + 61T^{2} \) |
| 67 | \( 1 - 4.97T + 67T^{2} \) |
| 71 | \( 1 - 0.840T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 3.36T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−7.71672711436719682063601504611, −6.82942496033014348771653996229, −6.20684533206937376657868830380, −5.27722519280650232216716378825, −4.92525096098511780037957904223, −3.48641904288138276072217843134, −3.22915052538416672294512313351, −2.05198506485032231518406313796, −1.10267600141168974818253999764, 0,
1.10267600141168974818253999764, 2.05198506485032231518406313796, 3.22915052538416672294512313351, 3.48641904288138276072217843134, 4.92525096098511780037957904223, 5.27722519280650232216716378825, 6.20684533206937376657868830380, 6.82942496033014348771653996229, 7.71672711436719682063601504611
