| L(s) = 1 | − 5.25·2-s + 3.13·3-s + 19.6·4-s − 5·5-s − 16.4·6-s − 24.7·7-s − 61.2·8-s − 17.1·9-s + 26.2·10-s + 17.6·11-s + 61.6·12-s − 13·13-s + 130.·14-s − 15.6·15-s + 164.·16-s − 95.7·17-s + 90.2·18-s + 115.·19-s − 98.2·20-s − 77.6·21-s − 92.7·22-s + 20.0·23-s − 192.·24-s + 25·25-s + 68.3·26-s − 138.·27-s − 486.·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 0.603·3-s + 2.45·4-s − 0.447·5-s − 1.12·6-s − 1.33·7-s − 2.70·8-s − 0.635·9-s + 0.831·10-s + 0.483·11-s + 1.48·12-s − 0.277·13-s + 2.48·14-s − 0.269·15-s + 2.57·16-s − 1.36·17-s + 1.18·18-s + 1.38·19-s − 1.09·20-s − 0.807·21-s − 0.898·22-s + 0.181·23-s − 1.63·24-s + 0.200·25-s + 0.515·26-s − 0.987·27-s − 3.28·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 5T \) |
| 13 | \( 1 + 13T \) |
| 31 | \( 1 - 31T \) |
| good | 2 | \( 1 + 5.25T + 8T^{2} \) |
| 3 | \( 1 - 3.13T + 27T^{2} \) |
| 7 | \( 1 + 24.7T + 343T^{2} \) |
| 11 | \( 1 - 17.6T + 1.33e3T^{2} \) |
| 17 | \( 1 + 95.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 115.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 20.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 88.6T + 2.43e4T^{2} \) |
| 37 | \( 1 + 266.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 446.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 151.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 273.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 470.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 227.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 77.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 656.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 335.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 789.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 763.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 778.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.18e3T + 9.12e5T^{2} \) |
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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
−8.575598512268378638982697866286, −7.87250611661756064293365920408, −6.97636893972987666872886734635, −6.62245098865290768144367256125, −5.54462958109087278592514971107, −3.84580019021362235519463194690, −2.94359141494310995639154274324, −2.31546905857335740705544015923, −0.887302188086556653088946501080, 0,
0.887302188086556653088946501080, 2.31546905857335740705544015923, 2.94359141494310995639154274324, 3.84580019021362235519463194690, 5.54462958109087278592514971107, 6.62245098865290768144367256125, 6.97636893972987666872886734635, 7.87250611661756064293365920408, 8.575598512268378638982697866286
