| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 3·7-s + 8-s + 9-s + 4·11-s − 12-s + 4·13-s − 3·14-s + 16-s + 5·17-s + 18-s + 2·19-s + 3·21-s + 4·22-s − 24-s + 4·26-s − 27-s − 3·28-s − 5·29-s + 2·31-s + 32-s − 4·33-s + 5·34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s + 1.10·13-s − 0.801·14-s + 1/4·16-s + 1.21·17-s + 0.235·18-s + 0.458·19-s + 0.654·21-s + 0.852·22-s − 0.204·24-s + 0.784·26-s − 0.192·27-s − 0.566·28-s − 0.928·29-s + 0.359·31-s + 0.176·32-s − 0.696·33-s + 0.857·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.130337377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.130337377\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.85675272387874, −13.57323035348721, −12.94768092664316, −12.46562845943042, −12.13395954365401, −11.64400700131047, −11.04236314657714, −10.72671878804964, −9.958098342152830, −9.539181095239445, −9.168560754381426, −8.376971262540964, −7.770102229193214, −7.041202225079405, −6.761190641850656, −6.092568990968134, −5.769274441716392, −5.352892854624910, −4.398460124544714, −3.893469986882248, −3.504496062662691, −2.952290137674141, −2.023548071594696, −1.175894461533014, −0.6989370818592091,
0.6989370818592091, 1.175894461533014, 2.023548071594696, 2.952290137674141, 3.504496062662691, 3.893469986882248, 4.398460124544714, 5.352892854624910, 5.769274441716392, 6.092568990968134, 6.761190641850656, 7.041202225079405, 7.770102229193214, 8.376971262540964, 9.168560754381426, 9.539181095239445, 9.958098342152830, 10.72671878804964, 11.04236314657714, 11.64400700131047, 12.13395954365401, 12.46562845943042, 12.94768092664316, 13.57323035348721, 13.85675272387874
