| L(s) = 1 | − 2-s − 2.59·3-s + 4-s + 1.58·5-s + 2.59·6-s + 7-s − 8-s + 3.70·9-s − 1.58·10-s + 5.04·11-s − 2.59·12-s − 2.41·13-s − 14-s − 4.10·15-s + 16-s − 2.62·17-s − 3.70·18-s − 4.64·19-s + 1.58·20-s − 2.59·21-s − 5.04·22-s − 5.45·23-s + 2.59·24-s − 2.48·25-s + 2.41·26-s − 1.83·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.49·3-s + 0.5·4-s + 0.708·5-s + 1.05·6-s + 0.377·7-s − 0.353·8-s + 1.23·9-s − 0.501·10-s + 1.52·11-s − 0.747·12-s − 0.669·13-s − 0.267·14-s − 1.05·15-s + 0.250·16-s − 0.637·17-s − 0.874·18-s − 1.06·19-s + 0.354·20-s − 0.565·21-s − 1.07·22-s − 1.13·23-s + 0.528·24-s − 0.497·25-s + 0.473·26-s − 0.353·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8203304873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8203304873\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 2.59T + 3T^{2} \) |
| 5 | \( 1 - 1.58T + 5T^{2} \) |
| 11 | \( 1 - 5.04T + 11T^{2} \) |
| 13 | \( 1 + 2.41T + 13T^{2} \) |
| 17 | \( 1 + 2.62T + 17T^{2} \) |
| 19 | \( 1 + 4.64T + 19T^{2} \) |
| 23 | \( 1 + 5.45T + 23T^{2} \) |
| 29 | \( 1 - 9.36T + 29T^{2} \) |
| 31 | \( 1 + 1.16T + 31T^{2} \) |
| 37 | \( 1 - 1.11T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 5.09T + 43T^{2} \) |
| 47 | \( 1 + 5.58T + 47T^{2} \) |
| 53 | \( 1 + 0.467T + 53T^{2} \) |
| 59 | \( 1 - 7.18T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 6.06T + 71T^{2} \) |
| 73 | \( 1 - 5.20T + 73T^{2} \) |
| 79 | \( 1 + 3.99T + 79T^{2} \) |
| 83 | \( 1 + 4.52T + 83T^{2} \) |
| 89 | \( 1 + 3.80T + 89T^{2} \) |
| 97 | \( 1 + 6.16T + 97T^{2} \) |
| 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
−8.243920546811808016001137355025, −7.05260042374148751701800498180, −6.54219721188278015426709270508, −6.20390282405583897327852659171, −5.36345711869604232466827007853, −4.61136982666876543411888202220, −3.85837378024833517915828945938, −2.31817930844961694524355920040, −1.63065182022519122662705000699, −0.58536989086008796571483775018,
0.58536989086008796571483775018, 1.63065182022519122662705000699, 2.31817930844961694524355920040, 3.85837378024833517915828945938, 4.61136982666876543411888202220, 5.36345711869604232466827007853, 6.20390282405583897327852659171, 6.54219721188278015426709270508, 7.05260042374148751701800498180, 8.243920546811808016001137355025
