| L(s) = 1 | + 2.42·2-s + 3-s + 3.88·4-s + 5-s + 2.42·6-s − 4.56·7-s + 4.57·8-s + 9-s + 2.42·10-s − 4.12·11-s + 3.88·12-s − 13-s − 11.0·14-s + 15-s + 3.32·16-s − 4.45·17-s + 2.42·18-s − 3.34·19-s + 3.88·20-s − 4.56·21-s − 9.99·22-s − 6.70·23-s + 4.57·24-s + 25-s − 2.42·26-s + 27-s − 17.7·28-s + ⋯ |
| L(s) = 1 | + 1.71·2-s + 0.577·3-s + 1.94·4-s + 0.447·5-s + 0.990·6-s − 1.72·7-s + 1.61·8-s + 0.333·9-s + 0.767·10-s − 1.24·11-s + 1.12·12-s − 0.277·13-s − 2.95·14-s + 0.258·15-s + 0.831·16-s − 1.07·17-s + 0.571·18-s − 0.767·19-s + 0.868·20-s − 0.995·21-s − 2.13·22-s − 1.39·23-s + 0.933·24-s + 0.200·25-s − 0.475·26-s + 0.192·27-s − 3.34·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 2.42T + 2T^{2} \) |
| 7 | \( 1 + 4.56T + 7T^{2} \) |
| 11 | \( 1 + 4.12T + 11T^{2} \) |
| 17 | \( 1 + 4.45T + 17T^{2} \) |
| 19 | \( 1 + 3.34T + 19T^{2} \) |
| 23 | \( 1 + 6.70T + 23T^{2} \) |
| 29 | \( 1 - 2.79T + 29T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 2.98T + 41T^{2} \) |
| 43 | \( 1 - 9.98T + 43T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 + 4.99T + 59T^{2} \) |
| 61 | \( 1 - 0.380T + 61T^{2} \) |
| 67 | \( 1 + 3.71T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 - 2.99T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 - 8.56T + 97T^{2} \) |
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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
−7.50755438628530927730928067891, −6.52638615691650776485084380748, −6.31706454947124696975426466736, −5.61614022004669438121494258703, −4.62612484257603768333307848895, −4.11480350066072214274626526196, −3.17938573734378262172768217219, −2.64098738662822787086599050901, −2.10313396306198347686131927498, 0,
2.10313396306198347686131927498, 2.64098738662822787086599050901, 3.17938573734378262172768217219, 4.11480350066072214274626526196, 4.62612484257603768333307848895, 5.61614022004669438121494258703, 6.31706454947124696975426466736, 6.52638615691650776485084380748, 7.50755438628530927730928067891
