| L(s) = 1 | − 3·3-s + 19.1·5-s + 9·9-s − 40.5·11-s − 50.4·13-s − 57.4·15-s + 51.9·17-s − 33.1·19-s − 62.8·23-s + 241.·25-s − 27·27-s + 129.·29-s + 242.·31-s + 121.·33-s − 389.·37-s + 151.·39-s + 470.·41-s + 125.·43-s + 172.·45-s − 386.·47-s − 155.·51-s − 611.·53-s − 777.·55-s + 99.3·57-s + 226.·59-s − 725.·61-s − 966.·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.71·5-s + 0.333·9-s − 1.11·11-s − 1.07·13-s − 0.988·15-s + 0.740·17-s − 0.400·19-s − 0.569·23-s + 1.93·25-s − 0.192·27-s + 0.831·29-s + 1.40·31-s + 0.642·33-s − 1.73·37-s + 0.621·39-s + 1.79·41-s + 0.443·43-s + 0.570·45-s − 1.19·47-s − 0.427·51-s − 1.58·53-s − 1.90·55-s + 0.230·57-s + 0.499·59-s − 1.52·61-s − 1.84·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 19.1T + 125T^{2} \) |
| 11 | \( 1 + 40.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 50.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 51.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 33.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 62.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 129.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 242.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 389.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 470.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 125.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 386.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 611.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 226.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 725.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.04e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 169.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 381.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 808.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 319.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.13e3T + 9.12e5T^{2} \) |
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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
−8.201963111303360398898644968998, −7.42351534503805486236298834781, −6.43522276674878757773363050909, −5.92682762225542793806737527423, −5.13162869377296296106993025227, −4.63390076871511002408591020317, −2.98427189217907215100627236013, −2.28537974750647339873794221488, −1.31068244076930782402594578819, 0,
1.31068244076930782402594578819, 2.28537974750647339873794221488, 2.98427189217907215100627236013, 4.63390076871511002408591020317, 5.13162869377296296106993025227, 5.92682762225542793806737527423, 6.43522276674878757773363050909, 7.42351534503805486236298834781, 8.201963111303360398898644968998
