| L(s) = 1 | + 2.08·2-s + 2.35·4-s + 5-s + 1.35·7-s + 0.734·8-s + 2.08·10-s − 3.73·11-s + 2.82·14-s − 3.17·16-s + 2.70·17-s − 0.438·19-s + 2.35·20-s − 7.79·22-s + 5.08·23-s + 25-s + 3.17·28-s + 1.35·29-s + 6.43·31-s − 8.08·32-s + 5.64·34-s + 1.35·35-s + 7.35·37-s − 0.913·38-s + 0.734·40-s + 6.87·41-s − 0.209·43-s − 8.78·44-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 1.17·4-s + 0.447·5-s + 0.510·7-s + 0.259·8-s + 0.659·10-s − 1.12·11-s + 0.753·14-s − 0.793·16-s + 0.655·17-s − 0.100·19-s + 0.525·20-s − 1.66·22-s + 1.06·23-s + 0.200·25-s + 0.600·28-s + 0.251·29-s + 1.15·31-s − 1.42·32-s + 0.967·34-s + 0.228·35-s + 1.20·37-s − 0.148·38-s + 0.116·40-s + 1.07·41-s − 0.0320·43-s − 1.32·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.296812875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.296812875\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.08T + 2T^{2} \) |
| 7 | \( 1 - 1.35T + 7T^{2} \) |
| 11 | \( 1 + 3.73T + 11T^{2} \) |
| 17 | \( 1 - 2.70T + 17T^{2} \) |
| 19 | \( 1 + 0.438T + 19T^{2} \) |
| 23 | \( 1 - 5.08T + 23T^{2} \) |
| 29 | \( 1 - 1.35T + 29T^{2} \) |
| 31 | \( 1 - 6.43T + 31T^{2} \) |
| 37 | \( 1 - 7.35T + 37T^{2} \) |
| 41 | \( 1 - 6.87T + 41T^{2} \) |
| 43 | \( 1 + 0.209T + 43T^{2} \) |
| 47 | \( 1 - 1.35T + 47T^{2} \) |
| 53 | \( 1 - 1.46T + 53T^{2} \) |
| 59 | \( 1 - 2.26T + 59T^{2} \) |
| 61 | \( 1 - 3.52T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 - 0.438T + 71T^{2} \) |
| 73 | \( 1 + 3.69T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + 0.475T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 + 3.29T + 97T^{2} \) |
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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
−7.77917483110804994145082224585, −6.95798664613708808027820693714, −6.22471198212272294022342027389, −5.60621613620994947119316024672, −4.98480760923770989490098305146, −4.56077780526234914857742073772, −3.58780797244453857109727515708, −2.77274157348718618554953751048, −2.26880825748809107091976063773, −0.930008591065161958034712359188,
0.930008591065161958034712359188, 2.26880825748809107091976063773, 2.77274157348718618554953751048, 3.58780797244453857109727515708, 4.56077780526234914857742073772, 4.98480760923770989490098305146, 5.60621613620994947119316024672, 6.22471198212272294022342027389, 6.95798664613708808027820693714, 7.77917483110804994145082224585
