| L(s) = 1 | − 2-s − 1.49·3-s + 4-s + 5-s + 1.49·6-s + 2.81·7-s − 8-s − 0.758·9-s − 10-s + 2.95·11-s − 1.49·12-s + 13-s − 2.81·14-s − 1.49·15-s + 16-s + 2.99·17-s + 0.758·18-s + 2.33·19-s + 20-s − 4.21·21-s − 2.95·22-s + 3.19·23-s + 1.49·24-s + 25-s − 26-s + 5.62·27-s + 2.81·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.864·3-s + 0.5·4-s + 0.447·5-s + 0.611·6-s + 1.06·7-s − 0.353·8-s − 0.252·9-s − 0.316·10-s + 0.892·11-s − 0.432·12-s + 0.277·13-s − 0.752·14-s − 0.386·15-s + 0.250·16-s + 0.726·17-s + 0.178·18-s + 0.535·19-s + 0.223·20-s − 0.919·21-s − 0.630·22-s + 0.665·23-s + 0.305·24-s + 0.200·25-s − 0.196·26-s + 1.08·27-s + 0.531·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.391472421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.391472421\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 + 1.49T + 3T^{2} \) |
| 7 | \( 1 - 2.81T + 7T^{2} \) |
| 11 | \( 1 - 2.95T + 11T^{2} \) |
| 17 | \( 1 - 2.99T + 17T^{2} \) |
| 19 | \( 1 - 2.33T + 19T^{2} \) |
| 23 | \( 1 - 3.19T + 23T^{2} \) |
| 29 | \( 1 + 2.38T + 29T^{2} \) |
| 37 | \( 1 + 7.48T + 37T^{2} \) |
| 41 | \( 1 - 6.76T + 41T^{2} \) |
| 43 | \( 1 - 5.25T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 - 4.39T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 + 6.95T + 67T^{2} \) |
| 71 | \( 1 + 8.34T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 + 7.66T + 79T^{2} \) |
| 83 | \( 1 + 0.750T + 83T^{2} \) |
| 89 | \( 1 - 0.992T + 89T^{2} \) |
| 97 | \( 1 - 5.46T + 97T^{2} \) |
| 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
−8.556956906407548114303621248588, −7.70811236463134804826303343662, −7.02142949399327779048079261990, −6.19726643862729343314833084852, −5.56257041456459333846856393078, −4.96540040185298727254596300472, −3.87140616778267367416176875235, −2.74508821539513553444947659872, −1.57763504954454169757007725618, −0.861172355167114485740385394380,
0.861172355167114485740385394380, 1.57763504954454169757007725618, 2.74508821539513553444947659872, 3.87140616778267367416176875235, 4.96540040185298727254596300472, 5.56257041456459333846856393078, 6.19726643862729343314833084852, 7.02142949399327779048079261990, 7.70811236463134804826303343662, 8.556956906407548114303621248588
