| L(s) = 1 | − 1.20·2-s − 0.545·4-s + 5-s − 4.31·7-s + 3.06·8-s − 1.20·10-s − 6.18·13-s + 5.20·14-s − 2.61·16-s + 0.803·17-s + 1.87·19-s − 0.545·20-s + 8.08·23-s + 25-s + 7.45·26-s + 2.35·28-s + 1.05·29-s − 1.61·31-s − 2.99·32-s − 0.969·34-s − 4.31·35-s − 5.54·37-s − 2.25·38-s + 3.06·40-s − 5.05·41-s + 9.87·43-s − 9.74·46-s + ⋯ |
| L(s) = 1 | − 0.852·2-s − 0.272·4-s + 0.447·5-s − 1.63·7-s + 1.08·8-s − 0.381·10-s − 1.71·13-s + 1.39·14-s − 0.652·16-s + 0.194·17-s + 0.429·19-s − 0.122·20-s + 1.68·23-s + 0.200·25-s + 1.46·26-s + 0.445·28-s + 0.196·29-s − 0.290·31-s − 0.528·32-s − 0.166·34-s − 0.729·35-s − 0.910·37-s − 0.366·38-s + 0.485·40-s − 0.788·41-s + 1.50·43-s − 1.43·46-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 | \( 1 + 1.20T + 2T^{2} \) |
| 7 | \( 1 + 4.31T + 7T^{2} \) |
| 13 | \( 1 + 6.18T + 13T^{2} \) |
| 17 | \( 1 - 0.803T + 17T^{2} \) |
| 19 | \( 1 - 1.87T + 19T^{2} \) |
| 23 | \( 1 - 8.08T + 23T^{2} \) |
| 29 | \( 1 - 1.05T + 29T^{2} \) |
| 31 | \( 1 + 1.61T + 31T^{2} \) |
| 37 | \( 1 + 5.54T + 37T^{2} \) |
| 41 | \( 1 + 5.05T + 41T^{2} \) |
| 43 | \( 1 - 9.87T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 + 2.66T + 53T^{2} \) |
| 59 | \( 1 - 6.60T + 59T^{2} \) |
| 61 | \( 1 - 5.34T + 61T^{2} \) |
| 67 | \( 1 + 7.49T + 67T^{2} \) |
| 71 | \( 1 + 5.82T + 71T^{2} \) |
| 73 | \( 1 + 4.80T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 1.34T + 97T^{2} \) |
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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
−7.71513009899718585785969695016, −7.15896874526133432302922347438, −6.69564257609370302506282591274, −5.57563652885978505688782737456, −5.04137272664577921937118220675, −4.06066118418531047817073474766, −3.07402001758763996458933035666, −2.37137752656288858130963650090, −1.00354310364536232533104100791, 0,
1.00354310364536232533104100791, 2.37137752656288858130963650090, 3.07402001758763996458933035666, 4.06066118418531047817073474766, 5.04137272664577921937118220675, 5.57563652885978505688782737456, 6.69564257609370302506282591274, 7.15896874526133432302922347438, 7.71513009899718585785969695016
