| L(s) = 1 | − 4.38·2-s + 3·3-s + 11.2·4-s − 10.2·5-s − 13.1·6-s + 22.1·7-s − 14.1·8-s + 9·9-s + 45.1·10-s + 11·11-s + 33.6·12-s + 16.1·13-s − 97.1·14-s − 30.8·15-s − 27.8·16-s + 26.3·17-s − 39.4·18-s + 76.7·19-s − 115.·20-s + 66.4·21-s − 48.2·22-s − 97.9·23-s − 42.3·24-s − 19.0·25-s − 70.6·26-s + 27·27-s + 248.·28-s + ⋯ |
| L(s) = 1 | − 1.55·2-s + 0.577·3-s + 1.40·4-s − 0.920·5-s − 0.894·6-s + 1.19·7-s − 0.623·8-s + 0.333·9-s + 1.42·10-s + 0.301·11-s + 0.809·12-s + 0.343·13-s − 1.85·14-s − 0.531·15-s − 0.435·16-s + 0.376·17-s − 0.516·18-s + 0.926·19-s − 1.29·20-s + 0.690·21-s − 0.467·22-s − 0.888·23-s − 0.360·24-s − 0.152·25-s − 0.532·26-s + 0.192·27-s + 1.67·28-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 | \( 1 + 4.38T + 8T^{2} \) |
| 5 | \( 1 + 10.2T + 125T^{2} \) |
| 7 | \( 1 - 22.1T + 343T^{2} \) |
| 13 | \( 1 - 16.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 26.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 76.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 97.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 17.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 42.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 279.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 54.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 89.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 492.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 685.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 628.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 1.02e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 38.6T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.12e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 471.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 880.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 394.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 84.0T + 9.12e5T^{2} \) |
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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
−8.175364900415171754403599580511, −7.996844710362471781119914642349, −7.39443497570403572456286150851, −6.44025049796344186171833224482, −5.10043497383244928048010257535, −4.17260197557901568907945533803, −3.21455203162654098632965282454, −1.88111634544781485315549332812, −1.23273189595762489011104348647, 0,
1.23273189595762489011104348647, 1.88111634544781485315549332812, 3.21455203162654098632965282454, 4.17260197557901568907945533803, 5.10043497383244928048010257535, 6.44025049796344186171833224482, 7.39443497570403572456286150851, 7.996844710362471781119914642349, 8.175364900415171754403599580511
