| L(s) = 1 | − 2.70·2-s + 5.34·4-s − 5-s − 1.07·7-s − 9.04·8-s + 2.70·10-s + 4.34·13-s + 2.92·14-s + 13.8·16-s + 7.75·17-s − 5.26·19-s − 5.34·20-s + 2.15·23-s + 25-s − 11.7·26-s − 5.75·28-s + 1.41·29-s − 4.68·31-s − 19.3·32-s − 21.0·34-s + 1.07·35-s − 2·37-s + 14.2·38-s + 9.04·40-s − 9.41·41-s − 7.60·43-s − 5.84·46-s + ⋯ |
| L(s) = 1 | − 1.91·2-s + 2.67·4-s − 0.447·5-s − 0.407·7-s − 3.19·8-s + 0.856·10-s + 1.20·13-s + 0.780·14-s + 3.45·16-s + 1.88·17-s − 1.20·19-s − 1.19·20-s + 0.449·23-s + 0.200·25-s − 2.30·26-s − 1.08·28-s + 0.263·29-s − 0.840·31-s − 3.42·32-s − 3.60·34-s + 0.182·35-s − 0.328·37-s + 2.31·38-s + 1.43·40-s − 1.47·41-s − 1.15·43-s − 0.861·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 7 | \( 1 + 1.07T + 7T^{2} \) |
| 13 | \( 1 - 4.34T + 13T^{2} \) |
| 17 | \( 1 - 7.75T + 17T^{2} \) |
| 19 | \( 1 + 5.26T + 19T^{2} \) |
| 23 | \( 1 - 2.15T + 23T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 + 4.68T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 9.41T + 41T^{2} \) |
| 43 | \( 1 + 7.60T + 43T^{2} \) |
| 47 | \( 1 + 4.68T + 47T^{2} \) |
| 53 | \( 1 + 0.156T + 53T^{2} \) |
| 59 | \( 1 + 6.15T + 59T^{2} \) |
| 61 | \( 1 - 4.15T + 61T^{2} \) |
| 67 | \( 1 + 8.68T + 67T^{2} \) |
| 71 | \( 1 - 4.68T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 8.09T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−8.058209446633277524341945918766, −7.34761624445102672646220656819, −6.57820513206009496686784736963, −6.13255691434261946194397147488, −5.12139338858819136365012137814, −3.56174705688401557978556857401, −3.20443516020982238814556841865, −1.90733479217931900555298096672, −1.12277345345335647661122554878, 0,
1.12277345345335647661122554878, 1.90733479217931900555298096672, 3.20443516020982238814556841865, 3.56174705688401557978556857401, 5.12139338858819136365012137814, 6.13255691434261946194397147488, 6.57820513206009496686784736963, 7.34761624445102672646220656819, 8.058209446633277524341945918766
