| L(s) = 1 | − 2-s − 3-s + 4-s + 2.82·5-s + 6-s − 2.82·7-s − 8-s + 9-s − 2.82·10-s − 5.65·11-s − 12-s + 6·13-s + 2.82·14-s − 2.82·15-s + 16-s − 2.82·17-s − 18-s − 8.48·19-s + 2.82·20-s + 2.82·21-s + 5.65·22-s + 24-s + 3.00·25-s − 6·26-s − 27-s − 2.82·28-s + 2·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.26·5-s + 0.408·6-s − 1.06·7-s − 0.353·8-s + 0.333·9-s − 0.894·10-s − 1.70·11-s − 0.288·12-s + 1.66·13-s + 0.755·14-s − 0.730·15-s + 0.250·16-s − 0.685·17-s − 0.235·18-s − 1.94·19-s + 0.632·20-s + 0.617·21-s + 1.20·22-s + 0.204·24-s + 0.600·25-s − 1.17·26-s − 0.192·27-s − 0.534·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9212061856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9212061856\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + 8.48T + 19T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 2.82T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 8.48T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 8.48T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 2.82T + 79T^{2} \) |
| 83 | \( 1 - 5.65T + 83T^{2} \) |
| 89 | \( 1 - 2.82T + 89T^{2} \) |
| 97 | \( 1 - 5.65T + 97T^{2} \) |
| 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
−8.613339723169085956538767060744, −8.231551877949842802380522915048, −6.92228417535080849797468466658, −6.31972796312762612983452117187, −5.98872826727581571987169413468, −5.10144193573603887853781343589, −3.92240830725493537619103468528, −2.68710113561051201311598099325, −2.04366982693407013162924073502, −0.63658041767195145781759221265,
0.63658041767195145781759221265, 2.04366982693407013162924073502, 2.68710113561051201311598099325, 3.92240830725493537619103468528, 5.10144193573603887853781343589, 5.98872826727581571987169413468, 6.31972796312762612983452117187, 6.92228417535080849797468466658, 8.231551877949842802380522915048, 8.613339723169085956538767060744
