| L(s) = 1 | − 2.56·2-s − 1.43·4-s − 0.561·5-s + 18.1·7-s + 24.1·8-s + 1.43·10-s − 64.7·11-s − 13·13-s − 46.5·14-s − 50.4·16-s + 25.5·17-s − 107.·19-s + 0.807·20-s + 165.·22-s − 73.2·23-s − 124.·25-s + 33.3·26-s − 26.1·28-s − 175.·29-s − 113.·31-s − 64.2·32-s − 65.4·34-s − 10.2·35-s + 114.·37-s + 276.·38-s − 13.5·40-s + 69.6·41-s + ⋯ |
| L(s) = 1 | − 0.905·2-s − 0.179·4-s − 0.0502·5-s + 0.981·7-s + 1.06·8-s + 0.0454·10-s − 1.77·11-s − 0.277·13-s − 0.888·14-s − 0.787·16-s + 0.364·17-s − 1.30·19-s + 0.00903·20-s + 1.60·22-s − 0.664·23-s − 0.997·25-s + 0.251·26-s − 0.176·28-s − 1.12·29-s − 0.655·31-s − 0.354·32-s − 0.330·34-s − 0.0492·35-s + 0.510·37-s + 1.18·38-s − 0.0536·40-s + 0.265·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 + 2.56T + 8T^{2} \) |
| 5 | \( 1 + 0.561T + 125T^{2} \) |
| 7 | \( 1 - 18.1T + 343T^{2} \) |
| 11 | \( 1 + 64.7T + 1.33e3T^{2} \) |
| 17 | \( 1 - 25.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 73.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 175.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 113.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 114.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 69.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 438.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 31.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 2.84T + 1.48e5T^{2} \) |
| 59 | \( 1 + 71.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 920.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 444.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 541.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 764.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 421.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 603.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.15e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 583.T + 9.12e5T^{2} \) |
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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
−12.58901836356098803197472894463, −11.06425850649763885701510454204, −10.43140371193838474581833723237, −9.286775519692529610059307449873, −8.018015394122309221217538723485, −7.66655583271470656933999512170, −5.56529669345699534903061496502, −4.36648649967123802853351447847, −2.05116396334526806829885490341, 0,
2.05116396334526806829885490341, 4.36648649967123802853351447847, 5.56529669345699534903061496502, 7.66655583271470656933999512170, 8.018015394122309221217538723485, 9.286775519692529610059307449873, 10.43140371193838474581833723237, 11.06425850649763885701510454204, 12.58901836356098803197472894463
