| L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s − 4·7-s − 8-s + 9-s + 2·10-s − 11-s + 12-s + 4·14-s − 2·15-s + 16-s − 18-s − 2·19-s − 2·20-s − 4·21-s + 22-s − 8·23-s − 24-s − 25-s + 27-s − 4·28-s + 2·29-s + 2·30-s − 10·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s + 0.288·12-s + 1.06·14-s − 0.516·15-s + 1/4·16-s − 0.235·18-s − 0.458·19-s − 0.447·20-s − 0.872·21-s + 0.213·22-s − 1.66·23-s − 0.204·24-s − 1/5·25-s + 0.192·27-s − 0.755·28-s + 0.371·29-s + 0.365·30-s − 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19074 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19074 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−16.16870442386158, −15.85707321556585, −15.30128524101219, −14.88839481119343, −14.08546125246666, −13.47258849121259, −12.97567946046681, −12.28404118795976, −12.00998515561616, −11.24326693679747, −10.50603640190534, −10.05304156595008, −9.586356912257428, −8.981282233979407, −8.226109042185117, −8.058779673551002, −7.043513016849918, −6.910972859435426, −6.046802827976339, −5.394858327797837, −4.250702764242029, −3.670313442066881, −3.230347082851201, −2.377669423144092, −1.590814209465111, 0, 0,
1.590814209465111, 2.377669423144092, 3.230347082851201, 3.670313442066881, 4.250702764242029, 5.394858327797837, 6.046802827976339, 6.910972859435426, 7.043513016849918, 8.058779673551002, 8.226109042185117, 8.981282233979407, 9.586356912257428, 10.05304156595008, 10.50603640190534, 11.24326693679747, 12.00998515561616, 12.28404118795976, 12.97567946046681, 13.47258849121259, 14.08546125246666, 14.88839481119343, 15.30128524101219, 15.85707321556585, 16.16870442386158
