| L(s) = 1 | + 0.845·2-s + 3-s − 1.28·4-s − 0.0676·5-s + 0.845·6-s − 2.72·7-s − 2.77·8-s + 9-s − 0.0571·10-s + 3.60·11-s − 1.28·12-s + 4.54·13-s − 2.30·14-s − 0.0676·15-s + 0.220·16-s + 17-s + 0.845·18-s − 7.93·19-s + 0.0868·20-s − 2.72·21-s + 3.04·22-s + 8.56·23-s − 2.77·24-s − 4.99·25-s + 3.84·26-s + 27-s + 3.49·28-s + ⋯ |
| L(s) = 1 | + 0.598·2-s + 0.577·3-s − 0.642·4-s − 0.0302·5-s + 0.345·6-s − 1.02·7-s − 0.982·8-s + 0.333·9-s − 0.0180·10-s + 1.08·11-s − 0.370·12-s + 1.26·13-s − 0.614·14-s − 0.0174·15-s + 0.0550·16-s + 0.242·17-s + 0.199·18-s − 1.82·19-s + 0.0194·20-s − 0.593·21-s + 0.649·22-s + 1.78·23-s − 0.567·24-s − 0.999·25-s + 0.754·26-s + 0.192·27-s + 0.660·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.428259203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.428259203\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 - 0.845T + 2T^{2} \) |
| 5 | \( 1 + 0.0676T + 5T^{2} \) |
| 7 | \( 1 + 2.72T + 7T^{2} \) |
| 11 | \( 1 - 3.60T + 11T^{2} \) |
| 13 | \( 1 - 4.54T + 13T^{2} \) |
| 19 | \( 1 + 7.93T + 19T^{2} \) |
| 23 | \( 1 - 8.56T + 23T^{2} \) |
| 29 | \( 1 + 2.86T + 29T^{2} \) |
| 31 | \( 1 + 1.58T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 + 3.37T + 41T^{2} \) |
| 43 | \( 1 + 4.88T + 43T^{2} \) |
| 47 | \( 1 + 4.53T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 - 6.87T + 59T^{2} \) |
| 61 | \( 1 - 7.55T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 1.26T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 5.41T + 79T^{2} \) |
| 83 | \( 1 - 1.96T + 83T^{2} \) |
| 89 | \( 1 + 5.81T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−8.020257795348480906480828568594, −6.89356642074463824582457045807, −6.33609471273751773906948121396, −5.92805495180538326310844015671, −4.82856226565267827182716579660, −4.11616582403180193244999122558, −3.56448793542742105666703387465, −3.07968088485290003456201948594, −1.87271997989418544433780035808, −0.69080043098148913679883653342,
0.69080043098148913679883653342, 1.87271997989418544433780035808, 3.07968088485290003456201948594, 3.56448793542742105666703387465, 4.11616582403180193244999122558, 4.82856226565267827182716579660, 5.92805495180538326310844015671, 6.33609471273751773906948121396, 6.89356642074463824582457045807, 8.020257795348480906480828568594
