| L(s) = 1 | − 2.54·2-s − 3-s + 4.46·4-s − 2.67·5-s + 2.54·6-s + 2.94·7-s − 6.26·8-s + 9-s + 6.80·10-s + 1.14·11-s − 4.46·12-s + 5.66·13-s − 7.49·14-s + 2.67·15-s + 7.00·16-s + 17-s − 2.54·18-s + 6.65·19-s − 11.9·20-s − 2.94·21-s − 2.92·22-s − 4.86·23-s + 6.26·24-s + 2.17·25-s − 14.4·26-s − 27-s + 13.1·28-s + ⋯ |
| L(s) = 1 | − 1.79·2-s − 0.577·3-s + 2.23·4-s − 1.19·5-s + 1.03·6-s + 1.11·7-s − 2.21·8-s + 0.333·9-s + 2.15·10-s + 0.346·11-s − 1.28·12-s + 1.57·13-s − 2.00·14-s + 0.691·15-s + 1.75·16-s + 0.242·17-s − 0.599·18-s + 1.52·19-s − 2.67·20-s − 0.642·21-s − 0.622·22-s − 1.01·23-s + 1.27·24-s + 0.434·25-s − 2.82·26-s − 0.192·27-s + 2.48·28-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
| \(L(1)\) |
\(\approx\) |
\(0.8883183289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8883183289\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.54T + 2T^{2} \) |
| 5 | \( 1 + 2.67T + 5T^{2} \) |
| 7 | \( 1 - 2.94T + 7T^{2} \) |
| 11 | \( 1 - 1.14T + 11T^{2} \) |
| 13 | \( 1 - 5.66T + 13T^{2} \) |
| 19 | \( 1 - 6.65T + 19T^{2} \) |
| 23 | \( 1 + 4.86T + 23T^{2} \) |
| 29 | \( 1 - 7.98T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 - 1.03T + 37T^{2} \) |
| 41 | \( 1 + 9.73T + 41T^{2} \) |
| 43 | \( 1 - 3.63T + 43T^{2} \) |
| 47 | \( 1 - 9.61T + 47T^{2} \) |
| 53 | \( 1 - 13.7T + 53T^{2} \) |
| 59 | \( 1 - 14.9T + 59T^{2} \) |
| 61 | \( 1 + 5.54T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 1.44T + 73T^{2} \) |
| 79 | \( 1 + 1.75T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 + 4.33T + 89T^{2} \) |
| 97 | \( 1 + 6.00T + 97T^{2} \) |
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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
−8.109032023764937548361350120852, −7.40299540096814020963653151107, −6.75692188642959481318573961423, −6.05395103285828247421237091296, −5.16210045023748137138634714378, −4.17695186084318687647572163132, −3.46470974540600265966943799873, −2.26638433556380903053097403900, −1.07721537266454228443128667245, −0.856401519873281767435337061776,
0.856401519873281767435337061776, 1.07721537266454228443128667245, 2.26638433556380903053097403900, 3.46470974540600265966943799873, 4.17695186084318687647572163132, 5.16210045023748137138634714378, 6.05395103285828247421237091296, 6.75692188642959481318573961423, 7.40299540096814020963653151107, 8.109032023764937548361350120852
