| L(s) = 1 | − 2-s + 3-s + 4-s − 3·5-s − 6-s − 3·7-s − 8-s + 9-s + 3·10-s + 2·11-s + 12-s + 3·14-s − 3·15-s + 16-s + 3·17-s − 18-s − 19-s − 3·20-s − 3·21-s − 2·22-s + 23-s − 24-s + 4·25-s + 27-s − 3·28-s − 29-s + 3·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.603·11-s + 0.288·12-s + 0.801·14-s − 0.774·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.229·19-s − 0.670·20-s − 0.654·21-s − 0.426·22-s + 0.208·23-s − 0.204·24-s + 4/5·25-s + 0.192·27-s − 0.566·28-s − 0.185·29-s + 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.181287385165343335154846329483, −7.51030725198775318183254953675, −6.77913959791946228009258145967, −6.25910920730510727427596292045, −4.97029829415090071042220572418, −3.88357020461952428072968382897, −3.43432195432228343033354965317, −2.60670009310683293711334821473, −1.18955940393403893990759813368, 0,
1.18955940393403893990759813368, 2.60670009310683293711334821473, 3.43432195432228343033354965317, 3.88357020461952428072968382897, 4.97029829415090071042220572418, 6.25910920730510727427596292045, 6.77913959791946228009258145967, 7.51030725198775318183254953675, 8.181287385165343335154846329483
