| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 2·7-s − 8-s + 9-s + 10-s + 12-s + 13-s + 2·14-s − 15-s + 16-s − 18-s − 2·19-s − 20-s − 2·21-s − 24-s + 25-s − 26-s + 27-s − 2·28-s + 30-s + 8·31-s − 32-s + 2·35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 0.277·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.458·19-s − 0.223·20-s − 0.436·21-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.377·28-s + 0.182·30-s + 1.43·31-s − 0.176·32-s + 0.338·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.741460507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.741460507\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.91492925359029, −12.58546303221226, −12.07209956109271, −11.64918589934890, −11.06338977660449, −10.48633040211558, −10.29905545034780, −9.638797031124132, −9.118299116789716, −8.958320647272481, −8.219865727698361, −7.913360863896358, −7.448531181111269, −6.874763484577447, −6.270043825353790, −6.158387333096229, −5.216333149102644, −4.627155230818684, −4.013067397060771, −3.531304144843502, −2.944338909388208, −2.490876742966620, −1.834878287071535, −1.021508672857513, −0.4588605927821511,
0.4588605927821511, 1.021508672857513, 1.834878287071535, 2.490876742966620, 2.944338909388208, 3.531304144843502, 4.013067397060771, 4.627155230818684, 5.216333149102644, 6.158387333096229, 6.270043825353790, 6.874763484577447, 7.448531181111269, 7.913360863896358, 8.219865727698361, 8.958320647272481, 9.118299116789716, 9.638797031124132, 10.29905545034780, 10.48633040211558, 11.06338977660449, 11.64918589934890, 12.07209956109271, 12.58546303221226, 12.91492925359029
