| L(s) = 1 | + 2.08·2-s + 3·3-s − 3.67·4-s − 5·5-s + 6.24·6-s + 15.0·7-s − 24.2·8-s + 9·9-s − 10.4·10-s − 4.52·11-s − 11.0·12-s − 46.0·13-s + 31.3·14-s − 15·15-s − 21.1·16-s − 29.2·17-s + 18.7·18-s − 69.5·19-s + 18.3·20-s + 45.1·21-s − 9.41·22-s − 97.2·23-s − 72.8·24-s + 25·25-s − 95.6·26-s + 27·27-s − 55.2·28-s + ⋯ |
| L(s) = 1 | + 0.735·2-s + 0.577·3-s − 0.459·4-s − 0.447·5-s + 0.424·6-s + 0.812·7-s − 1.07·8-s + 0.333·9-s − 0.328·10-s − 0.124·11-s − 0.265·12-s − 0.981·13-s + 0.597·14-s − 0.258·15-s − 0.330·16-s − 0.417·17-s + 0.245·18-s − 0.840·19-s + 0.205·20-s + 0.469·21-s − 0.0912·22-s − 0.881·23-s − 0.619·24-s + 0.200·25-s − 0.721·26-s + 0.192·27-s − 0.373·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 2.08T + 8T^{2} \) |
| 7 | \( 1 - 15.0T + 343T^{2} \) |
| 11 | \( 1 + 4.52T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 29.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 69.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 97.2T + 1.21e4T^{2} \) |
| 31 | \( 1 - 28.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 423.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 299.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 302.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 7.16T + 1.03e5T^{2} \) |
| 53 | \( 1 + 602.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 139.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 866.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 823.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 201.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.10e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 89.3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 589.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 926.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 840.T + 9.12e5T^{2} \) |
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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
−10.25823493904588647297680732518, −9.240542487176694155903004741543, −8.394982498842627627767358110270, −7.66346297784182746153810786804, −6.41128177053884121388231687536, −5.05820479491504170794441593692, −4.43643196139673709859606233206, −3.38457364206971820234017084145, −2.05910758858564424250780288909, 0,
2.05910758858564424250780288909, 3.38457364206971820234017084145, 4.43643196139673709859606233206, 5.05820479491504170794441593692, 6.41128177053884121388231687536, 7.66346297784182746153810786804, 8.394982498842627627767358110270, 9.240542487176694155903004741543, 10.25823493904588647297680732518
