| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s + 7-s − 8-s + 9-s − 2·10-s − 12-s − 14-s − 2·15-s + 16-s + 17-s − 18-s + 2·20-s − 21-s − 8·23-s + 24-s − 25-s − 27-s + 28-s − 6·29-s + 2·30-s + 8·31-s − 32-s − 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s − 0.267·14-s − 0.516·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.447·20-s − 0.218·21-s − 1.66·23-s + 0.204·24-s − 1/5·25-s − 0.192·27-s + 0.188·28-s − 1.11·29-s + 0.365·30-s + 1.43·31-s − 0.176·32-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.93951397133886, −13.29198362956171, −12.70618760704740, −12.31738740344881, −11.69583795201320, −11.44034343354277, −10.76364522084019, −10.30618750345257, −9.999096039602882, −9.455305520811485, −9.068165034151341, −8.310090083992830, −7.986432423952770, −7.339204515348772, −6.911825001545416, −6.183236780167041, −5.739356042981268, −5.606899274534464, −4.660492059224205, −4.186805924260573, −3.472395227478202, −2.636442276325085, −2.020459964950944, −1.612789438475070, −0.8331701865992436, 0,
0.8331701865992436, 1.612789438475070, 2.020459964950944, 2.636442276325085, 3.472395227478202, 4.186805924260573, 4.660492059224205, 5.606899274534464, 5.739356042981268, 6.183236780167041, 6.911825001545416, 7.339204515348772, 7.986432423952770, 8.310090083992830, 9.068165034151341, 9.455305520811485, 9.999096039602882, 10.30618750345257, 10.76364522084019, 11.44034343354277, 11.69583795201320, 12.31738740344881, 12.70618760704740, 13.29198362956171, 13.93951397133886
