| L(s) = 1 | − 2-s − 3-s + 4-s + 3.46·5-s + 6-s + 4.73·7-s − 8-s + 9-s − 3.46·10-s − 2·11-s − 12-s − 2.73·13-s − 4.73·14-s − 3.46·15-s + 16-s − 3.46·17-s − 18-s + 2·19-s + 3.46·20-s − 4.73·21-s + 2·22-s + 8.73·23-s + 24-s + 6.99·25-s + 2.73·26-s − 27-s + 4.73·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.54·5-s + 0.408·6-s + 1.78·7-s − 0.353·8-s + 0.333·9-s − 1.09·10-s − 0.603·11-s − 0.288·12-s − 0.757·13-s − 1.26·14-s − 0.894·15-s + 0.250·16-s − 0.840·17-s − 0.235·18-s + 0.458·19-s + 0.774·20-s − 1.03·21-s + 0.426·22-s + 1.82·23-s + 0.204·24-s + 1.39·25-s + 0.535·26-s − 0.192·27-s + 0.894·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 402 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 402 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.228409778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.228409778\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 - 4.73T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 8.73T + 23T^{2} \) |
| 29 | \( 1 + 9.66T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 5.46T + 41T^{2} \) |
| 43 | \( 1 + 2.92T + 43T^{2} \) |
| 47 | \( 1 + 7.66T + 47T^{2} \) |
| 53 | \( 1 - 7.46T + 53T^{2} \) |
| 59 | \( 1 + 2.53T + 59T^{2} \) |
| 61 | \( 1 + 1.26T + 61T^{2} \) |
| 71 | \( 1 + 2.19T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 2.19T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
| 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
−11.08068374029420581092241606973, −10.39776287786352935461953601128, −9.528523274669312518192304251317, −8.628808165287054959086142970561, −7.56860924882028502053950743285, −6.61452452918885347607687009306, −5.32941933055802935429599743696, −4.92563890177238474253783309161, −2.43828504378367114783290923465, −1.44092575966250046242464974043,
1.44092575966250046242464974043, 2.43828504378367114783290923465, 4.92563890177238474253783309161, 5.32941933055802935429599743696, 6.61452452918885347607687009306, 7.56860924882028502053950743285, 8.628808165287054959086142970561, 9.528523274669312518192304251317, 10.39776287786352935461953601128, 11.08068374029420581092241606973
