| L(s) = 1 | − 2-s + 1.52·3-s + 4-s + 1.70·5-s − 1.52·6-s + 3.78·7-s − 8-s − 0.677·9-s − 1.70·10-s + 1.23·11-s + 1.52·12-s + 4.71·13-s − 3.78·14-s + 2.60·15-s + 16-s − 0.289·17-s + 0.677·18-s − 1.30·19-s + 1.70·20-s + 5.77·21-s − 1.23·22-s − 23-s − 1.52·24-s − 2.08·25-s − 4.71·26-s − 5.60·27-s + 3.78·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.879·3-s + 0.5·4-s + 0.763·5-s − 0.622·6-s + 1.43·7-s − 0.353·8-s − 0.225·9-s − 0.539·10-s + 0.371·11-s + 0.439·12-s + 1.30·13-s − 1.01·14-s + 0.671·15-s + 0.250·16-s − 0.0701·17-s + 0.159·18-s − 0.300·19-s + 0.381·20-s + 1.26·21-s − 0.262·22-s − 0.208·23-s − 0.311·24-s − 0.416·25-s − 0.924·26-s − 1.07·27-s + 0.716·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.955626955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.955626955\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 1.52T + 3T^{2} \) |
| 5 | \( 1 - 1.70T + 5T^{2} \) |
| 7 | \( 1 - 3.78T + 7T^{2} \) |
| 11 | \( 1 - 1.23T + 11T^{2} \) |
| 13 | \( 1 - 4.71T + 13T^{2} \) |
| 17 | \( 1 + 0.289T + 17T^{2} \) |
| 19 | \( 1 + 1.30T + 19T^{2} \) |
| 29 | \( 1 + 7.94T + 29T^{2} \) |
| 31 | \( 1 - 0.275T + 31T^{2} \) |
| 37 | \( 1 - 3.06T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 - 2.85T + 43T^{2} \) |
| 47 | \( 1 + 0.217T + 47T^{2} \) |
| 53 | \( 1 + 3.21T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 - 8.71T + 67T^{2} \) |
| 71 | \( 1 - 0.445T + 71T^{2} \) |
| 73 | \( 1 + 1.59T + 73T^{2} \) |
| 79 | \( 1 - 7.26T + 79T^{2} \) |
| 83 | \( 1 + 4.82T + 83T^{2} \) |
| 89 | \( 1 - 4.46T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
| 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
−8.119385930470092079183414227779, −7.76873009178819271744274141097, −6.80483631853385003074565448572, −5.85924730992231704449155917478, −5.51175851851143537448286687440, −4.22923108286821208999470832393, −3.59478438158316997435178786930, −2.36103964720430279192322683277, −1.94156137292005661806109756458, −1.02214097630685778741782779707,
1.02214097630685778741782779707, 1.94156137292005661806109756458, 2.36103964720430279192322683277, 3.59478438158316997435178786930, 4.22923108286821208999470832393, 5.51175851851143537448286687440, 5.85924730992231704449155917478, 6.80483631853385003074565448572, 7.76873009178819271744274141097, 8.119385930470092079183414227779
