| L(s) = 1 | − 2.71·2-s − 3.40·3-s + 5.39·4-s + 0.859·5-s + 9.27·6-s + 3.37·7-s − 9.24·8-s + 8.62·9-s − 2.33·10-s + 2.40·11-s − 18.4·12-s − 3.48·13-s − 9.16·14-s − 2.92·15-s + 14.3·16-s + 0.684·17-s − 23.4·18-s − 7.35·19-s + 4.63·20-s − 11.4·21-s − 6.54·22-s − 0.186·23-s + 31.5·24-s − 4.26·25-s + 9.47·26-s − 19.1·27-s + 18.1·28-s + ⋯ |
| L(s) = 1 | − 1.92·2-s − 1.96·3-s + 2.69·4-s + 0.384·5-s + 3.78·6-s + 1.27·7-s − 3.26·8-s + 2.87·9-s − 0.738·10-s + 0.725·11-s − 5.31·12-s − 0.966·13-s − 2.45·14-s − 0.756·15-s + 3.58·16-s + 0.165·17-s − 5.52·18-s − 1.68·19-s + 1.03·20-s − 2.50·21-s − 1.39·22-s − 0.0389·23-s + 6.43·24-s − 0.852·25-s + 1.85·26-s − 3.69·27-s + 3.43·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 53 | \( 1 + T \) |
| 151 | \( 1 + T \) |
| good | 2 | \( 1 + 2.71T + 2T^{2} \) |
| 3 | \( 1 + 3.40T + 3T^{2} \) |
| 5 | \( 1 - 0.859T + 5T^{2} \) |
| 7 | \( 1 - 3.37T + 7T^{2} \) |
| 11 | \( 1 - 2.40T + 11T^{2} \) |
| 13 | \( 1 + 3.48T + 13T^{2} \) |
| 17 | \( 1 - 0.684T + 17T^{2} \) |
| 19 | \( 1 + 7.35T + 19T^{2} \) |
| 23 | \( 1 + 0.186T + 23T^{2} \) |
| 29 | \( 1 - 1.47T + 29T^{2} \) |
| 31 | \( 1 - 7.64T + 31T^{2} \) |
| 37 | \( 1 - 3.39T + 37T^{2} \) |
| 41 | \( 1 - 3.74T + 41T^{2} \) |
| 43 | \( 1 - 5.35T + 43T^{2} \) |
| 47 | \( 1 - 5.09T + 47T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 - 7.11T + 71T^{2} \) |
| 73 | \( 1 + 0.544T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 4.02T + 83T^{2} \) |
| 89 | \( 1 - 4.74T + 89T^{2} \) |
| 97 | \( 1 + 8.94T + 97T^{2} \) |
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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
−7.70031925616669727421170557862, −6.79166231985879129480770319054, −6.25763100749447627454650059389, −5.84396565812243528718411720977, −4.81362407842260072965042024251, −4.23009045355245341108544826451, −2.38859743079844657270412289497, −1.67515923314323385517804211065, −1.00178290635005436577593308137, 0,
1.00178290635005436577593308137, 1.67515923314323385517804211065, 2.38859743079844657270412289497, 4.23009045355245341108544826451, 4.81362407842260072965042024251, 5.84396565812243528718411720977, 6.25763100749447627454650059389, 6.79166231985879129480770319054, 7.70031925616669727421170557862
