| L(s) = 1 | + 2-s + 4-s − 4.04·5-s + 0.692·7-s + 8-s − 4.04·10-s + 4.85·11-s + 0.692·14-s + 16-s − 7.38·17-s − 1.78·19-s − 4.04·20-s + 4.85·22-s − 5.10·23-s + 11.3·25-s + 0.692·28-s + 3.34·29-s + 0.972·31-s + 32-s − 7.38·34-s − 2.80·35-s + 1.28·37-s − 1.78·38-s − 4.04·40-s − 1.50·41-s − 8.31·43-s + 4.85·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.81·5-s + 0.261·7-s + 0.353·8-s − 1.28·10-s + 1.46·11-s + 0.184·14-s + 0.250·16-s − 1.79·17-s − 0.408·19-s − 0.905·20-s + 1.03·22-s − 1.06·23-s + 2.27·25-s + 0.130·28-s + 0.621·29-s + 0.174·31-s + 0.176·32-s − 1.26·34-s − 0.473·35-s + 0.211·37-s − 0.288·38-s − 0.640·40-s − 0.235·41-s − 1.26·43-s + 0.731·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 4.04T + 5T^{2} \) |
| 7 | \( 1 - 0.692T + 7T^{2} \) |
| 11 | \( 1 - 4.85T + 11T^{2} \) |
| 17 | \( 1 + 7.38T + 17T^{2} \) |
| 19 | \( 1 + 1.78T + 19T^{2} \) |
| 23 | \( 1 + 5.10T + 23T^{2} \) |
| 29 | \( 1 - 3.34T + 29T^{2} \) |
| 31 | \( 1 - 0.972T + 31T^{2} \) |
| 37 | \( 1 - 1.28T + 37T^{2} \) |
| 41 | \( 1 + 1.50T + 41T^{2} \) |
| 43 | \( 1 + 8.31T + 43T^{2} \) |
| 47 | \( 1 - 7.20T + 47T^{2} \) |
| 53 | \( 1 + 13.4T + 53T^{2} \) |
| 59 | \( 1 + 1.30T + 59T^{2} \) |
| 61 | \( 1 + 0.396T + 61T^{2} \) |
| 67 | \( 1 - 6.05T + 67T^{2} \) |
| 71 | \( 1 - 1.32T + 71T^{2} \) |
| 73 | \( 1 + 7.65T + 73T^{2} \) |
| 79 | \( 1 + 8.33T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 + 3.10T + 89T^{2} \) |
| 97 | \( 1 + 8.54T + 97T^{2} \) |
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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
−8.361857692340964722875491255891, −7.50719277773770057838819113870, −6.71177688853144678335199103810, −6.31265329841930805232147794356, −4.88771208837074146954305591760, −4.23204533733098603904573318429, −3.90681291207331497293349134746, −2.86887352156986316756235186105, −1.56759099700760226058139542948, 0,
1.56759099700760226058139542948, 2.86887352156986316756235186105, 3.90681291207331497293349134746, 4.23204533733098603904573318429, 4.88771208837074146954305591760, 6.31265329841930805232147794356, 6.71177688853144678335199103810, 7.50719277773770057838819113870, 8.361857692340964722875491255891
