| L(s) = 1 | − 2-s + 3-s + 4-s − 0.356·5-s − 6-s − 4.04·7-s − 8-s + 9-s + 0.356·10-s + 0.911·11-s + 12-s + 4.04·14-s − 0.356·15-s + 16-s − 2.09·17-s − 18-s + 4.98·19-s − 0.356·20-s − 4.04·21-s − 0.911·22-s + 8.49·23-s − 24-s − 4.87·25-s + 27-s − 4.04·28-s + 8.51·29-s + 0.356·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.159·5-s − 0.408·6-s − 1.53·7-s − 0.353·8-s + 0.333·9-s + 0.112·10-s + 0.274·11-s + 0.288·12-s + 1.08·14-s − 0.0921·15-s + 0.250·16-s − 0.508·17-s − 0.235·18-s + 1.14·19-s − 0.0798·20-s − 0.883·21-s − 0.194·22-s + 1.77·23-s − 0.204·24-s − 0.974·25-s + 0.192·27-s − 0.765·28-s + 1.58·29-s + 0.0651·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.187794774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.187794774\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 0.356T + 5T^{2} \) |
| 7 | \( 1 + 4.04T + 7T^{2} \) |
| 11 | \( 1 - 0.911T + 11T^{2} \) |
| 17 | \( 1 + 2.09T + 17T^{2} \) |
| 19 | \( 1 - 4.98T + 19T^{2} \) |
| 23 | \( 1 - 8.49T + 23T^{2} \) |
| 29 | \( 1 - 8.51T + 29T^{2} \) |
| 31 | \( 1 - 10.7T + 31T^{2} \) |
| 37 | \( 1 - 0.615T + 37T^{2} \) |
| 41 | \( 1 - 7.60T + 41T^{2} \) |
| 43 | \( 1 + 6.27T + 43T^{2} \) |
| 47 | \( 1 + 1.78T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 6.04T + 59T^{2} \) |
| 61 | \( 1 + 3.10T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 0.533T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 6.49T + 83T^{2} \) |
| 89 | \( 1 - 6.49T + 89T^{2} \) |
| 97 | \( 1 - 1.96T + 97T^{2} \) |
| 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
−9.870824776878324751252205154115, −9.135311224362448943950807052588, −8.515946485045853002081019226149, −7.43477749687803727867578798690, −6.77834408758778658600133648616, −6.00421437108238013889510175075, −4.55599552114216611354408033010, −3.27963387769484010305639416499, −2.69595240870254514358033529224, −0.925337255659619455615447351878,
0.925337255659619455615447351878, 2.69595240870254514358033529224, 3.27963387769484010305639416499, 4.55599552114216611354408033010, 6.00421437108238013889510175075, 6.77834408758778658600133648616, 7.43477749687803727867578798690, 8.515946485045853002081019226149, 9.135311224362448943950807052588, 9.870824776878324751252205154115
