| L(s) = 1 | − 4.85·2-s − 3·3-s + 15.5·4-s − 6.76·5-s + 14.5·6-s − 14.4·7-s − 36.5·8-s + 9·9-s + 32.8·10-s + 11·11-s − 46.5·12-s + 89.5·13-s + 69.9·14-s + 20.3·15-s + 52.9·16-s + 75.6·17-s − 43.6·18-s − 53.1·19-s − 105.·20-s + 43.2·21-s − 53.3·22-s − 60.1·23-s + 109.·24-s − 79.1·25-s − 434.·26-s − 27·27-s − 224.·28-s + ⋯ |
| L(s) = 1 | − 1.71·2-s − 0.577·3-s + 1.94·4-s − 0.605·5-s + 0.990·6-s − 0.778·7-s − 1.61·8-s + 0.333·9-s + 1.03·10-s + 0.301·11-s − 1.12·12-s + 1.90·13-s + 1.33·14-s + 0.349·15-s + 0.828·16-s + 1.07·17-s − 0.571·18-s − 0.642·19-s − 1.17·20-s + 0.449·21-s − 0.517·22-s − 0.545·23-s + 0.932·24-s − 0.633·25-s − 3.27·26-s − 0.192·27-s − 1.51·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 + 4.85T + 8T^{2} \) |
| 5 | \( 1 + 6.76T + 125T^{2} \) |
| 7 | \( 1 + 14.4T + 343T^{2} \) |
| 13 | \( 1 - 89.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 75.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 53.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 60.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 99.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 238.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 17.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 140.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 4.39T + 7.95e4T^{2} \) |
| 47 | \( 1 + 582.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 2.72T + 1.48e5T^{2} \) |
| 59 | \( 1 - 518.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 347.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 563.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 392.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 180.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 139.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 438.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.45e3T + 9.12e5T^{2} \) |
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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
−8.287184168490970392753953711901, −7.983925517876930378353520549565, −6.88103269907024082968268613973, −6.38843698422890168617217351777, −5.60688188710297584444676813544, −4.02612405245494636079664828788, −3.31737804785599036270337369316, −1.81496549496001021463596993070, −0.912223534813664490507189915184, 0,
0.912223534813664490507189915184, 1.81496549496001021463596993070, 3.31737804785599036270337369316, 4.02612405245494636079664828788, 5.60688188710297584444676813544, 6.38843698422890168617217351777, 6.88103269907024082968268613973, 7.983925517876930378353520549565, 8.287184168490970392753953711901
