| L(s) = 1 | + 2.16·2-s − 1.55·3-s + 2.69·4-s + 5-s − 3.36·6-s + 1.94·7-s + 1.50·8-s − 0.592·9-s + 2.16·10-s + 11-s − 4.18·12-s + 4.81·13-s + 4.22·14-s − 1.55·15-s − 2.12·16-s + 4.21·17-s − 1.28·18-s − 4.40·19-s + 2.69·20-s − 3.02·21-s + 2.16·22-s − 1.88·23-s − 2.33·24-s + 25-s + 10.4·26-s + 5.57·27-s + 5.25·28-s + ⋯ |
| L(s) = 1 | + 1.53·2-s − 0.895·3-s + 1.34·4-s + 0.447·5-s − 1.37·6-s + 0.737·7-s + 0.532·8-s − 0.197·9-s + 0.685·10-s + 0.301·11-s − 1.20·12-s + 1.33·13-s + 1.12·14-s − 0.400·15-s − 0.531·16-s + 1.02·17-s − 0.302·18-s − 1.01·19-s + 0.602·20-s − 0.660·21-s + 0.461·22-s − 0.392·23-s − 0.476·24-s + 0.200·25-s + 2.04·26-s + 1.07·27-s + 0.993·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.108454554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.108454554\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 - 2.16T + 2T^{2} \) |
| 3 | \( 1 + 1.55T + 3T^{2} \) |
| 7 | \( 1 - 1.94T + 7T^{2} \) |
| 13 | \( 1 - 4.81T + 13T^{2} \) |
| 17 | \( 1 - 4.21T + 17T^{2} \) |
| 19 | \( 1 + 4.40T + 19T^{2} \) |
| 23 | \( 1 + 1.88T + 23T^{2} \) |
| 29 | \( 1 - 7.25T + 29T^{2} \) |
| 31 | \( 1 + 7.65T + 31T^{2} \) |
| 37 | \( 1 - 8.64T + 37T^{2} \) |
| 41 | \( 1 + 2.23T + 41T^{2} \) |
| 43 | \( 1 - 9.48T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 3.93T + 53T^{2} \) |
| 59 | \( 1 - 1.70T + 59T^{2} \) |
| 61 | \( 1 - 4.40T + 61T^{2} \) |
| 67 | \( 1 + 2.87T + 67T^{2} \) |
| 71 | \( 1 + 6.41T + 71T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 - 0.759T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−8.464967017662376350243874643472, −7.41220447458418379234881332079, −6.48411280609378224628563100192, −5.80299173588672433497871883727, −5.72710581945276805356611914484, −4.67674129349942895395556782041, −4.12832851355355777165518493337, −3.19433119245943952508325058399, −2.18372797222680687606226940247, −1.01354274993120095396128963810,
1.01354274993120095396128963810, 2.18372797222680687606226940247, 3.19433119245943952508325058399, 4.12832851355355777165518493337, 4.67674129349942895395556782041, 5.72710581945276805356611914484, 5.80299173588672433497871883727, 6.48411280609378224628563100192, 7.41220447458418379234881332079, 8.464967017662376350243874643472
