| L(s) = 1 | + 0.560·2-s − 0.387·3-s − 1.68·4-s − 5-s − 0.217·6-s − 7-s − 2.06·8-s − 2.85·9-s − 0.560·10-s + 2.00·11-s + 0.652·12-s + 0.0912·13-s − 0.560·14-s + 0.387·15-s + 2.21·16-s − 0.0718·17-s − 1.59·18-s + 0.857·19-s + 1.68·20-s + 0.387·21-s + 1.12·22-s + 2.47·23-s + 0.799·24-s + 25-s + 0.0511·26-s + 2.26·27-s + 1.68·28-s + ⋯ |
| L(s) = 1 | + 0.396·2-s − 0.223·3-s − 0.842·4-s − 0.447·5-s − 0.0886·6-s − 0.377·7-s − 0.730·8-s − 0.950·9-s − 0.177·10-s + 0.604·11-s + 0.188·12-s + 0.0253·13-s − 0.149·14-s + 0.0999·15-s + 0.553·16-s − 0.0174·17-s − 0.376·18-s + 0.196·19-s + 0.376·20-s + 0.0844·21-s + 0.239·22-s + 0.515·23-s + 0.163·24-s + 0.200·25-s + 0.0100·26-s + 0.435·27-s + 0.318·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 - 0.560T + 2T^{2} \) |
| 3 | \( 1 + 0.387T + 3T^{2} \) |
| 11 | \( 1 - 2.00T + 11T^{2} \) |
| 13 | \( 1 - 0.0912T + 13T^{2} \) |
| 17 | \( 1 + 0.0718T + 17T^{2} \) |
| 19 | \( 1 - 0.857T + 19T^{2} \) |
| 23 | \( 1 - 2.47T + 23T^{2} \) |
| 29 | \( 1 + 3.66T + 29T^{2} \) |
| 31 | \( 1 + 0.585T + 31T^{2} \) |
| 37 | \( 1 - 4.76T + 37T^{2} \) |
| 41 | \( 1 + 8.94T + 41T^{2} \) |
| 43 | \( 1 - 6.06T + 43T^{2} \) |
| 47 | \( 1 + 6.90T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 7.99T + 59T^{2} \) |
| 61 | \( 1 + 3.96T + 61T^{2} \) |
| 67 | \( 1 + 4.32T + 67T^{2} \) |
| 71 | \( 1 - 9.47T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 - 7.11T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 + 6.61T + 97T^{2} \) |
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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
−7.56595893043395531477207733073, −6.56933858229819196136565104406, −6.08631595518783103783470753983, −5.25317423786348995492426845910, −4.78127222864432711348969600232, −3.76338143739388532862598168142, −3.42589190995727299531419966017, −2.45071105383465161597003768391, −0.986774652056342842474277477661, 0,
0.986774652056342842474277477661, 2.45071105383465161597003768391, 3.42589190995727299531419966017, 3.76338143739388532862598168142, 4.78127222864432711348969600232, 5.25317423786348995492426845910, 6.08631595518783103783470753983, 6.56933858229819196136565104406, 7.56595893043395531477207733073
