| L(s) = 1 | − 0.172·2-s + 3-s − 1.97·4-s + 1.16·5-s − 0.172·6-s − 2.85·7-s + 0.686·8-s + 9-s − 0.201·10-s + 4.87·11-s − 1.97·12-s − 5.45·13-s + 0.493·14-s + 1.16·15-s + 3.82·16-s + 17-s − 0.172·18-s + 1.41·19-s − 2.28·20-s − 2.85·21-s − 0.844·22-s + 2.01·23-s + 0.686·24-s − 3.64·25-s + 0.943·26-s + 27-s + 5.62·28-s + ⋯ |
| L(s) = 1 | − 0.122·2-s + 0.577·3-s − 0.985·4-s + 0.519·5-s − 0.0706·6-s − 1.07·7-s + 0.242·8-s + 0.333·9-s − 0.0635·10-s + 1.47·11-s − 0.568·12-s − 1.51·13-s + 0.131·14-s + 0.300·15-s + 0.955·16-s + 0.242·17-s − 0.0407·18-s + 0.323·19-s − 0.512·20-s − 0.622·21-s − 0.179·22-s + 0.419·23-s + 0.140·24-s − 0.729·25-s + 0.185·26-s + 0.192·27-s + 1.06·28-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
| \(L(1)\) |
\(\approx\) |
\(1.649650751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.649650751\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.172T + 2T^{2} \) |
| 5 | \( 1 - 1.16T + 5T^{2} \) |
| 7 | \( 1 + 2.85T + 7T^{2} \) |
| 11 | \( 1 - 4.87T + 11T^{2} \) |
| 13 | \( 1 + 5.45T + 13T^{2} \) |
| 19 | \( 1 - 1.41T + 19T^{2} \) |
| 23 | \( 1 - 2.01T + 23T^{2} \) |
| 29 | \( 1 + 3.80T + 29T^{2} \) |
| 31 | \( 1 - 3.10T + 31T^{2} \) |
| 37 | \( 1 + 0.374T + 37T^{2} \) |
| 41 | \( 1 - 2.70T + 41T^{2} \) |
| 43 | \( 1 - 1.72T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 7.42T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 + 0.930T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 + 0.00246T + 73T^{2} \) |
| 79 | \( 1 + 4.70T + 79T^{2} \) |
| 83 | \( 1 - 6.05T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 - 3.55T + 97T^{2} \) |
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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
−7.80786147377859540647225368521, −7.24499496903805680977646034284, −6.46900644754978152220861247126, −5.78711616470081023388385398904, −4.95422019707894622589003130607, −4.17965881106122293024222225302, −3.55200815523016988988336488832, −2.77390497893274720763403331335, −1.75347093005198841988635550972, −0.63904682843582752887671791288,
0.63904682843582752887671791288, 1.75347093005198841988635550972, 2.77390497893274720763403331335, 3.55200815523016988988336488832, 4.17965881106122293024222225302, 4.95422019707894622589003130607, 5.78711616470081023388385398904, 6.46900644754978152220861247126, 7.24499496903805680977646034284, 7.80786147377859540647225368521
