| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 12.1·5-s + 6·6-s + 7·7-s + 8·8-s + 9·9-s + 24.2·10-s + 10.1·11-s + 12·12-s − 2.78·13-s + 14·14-s + 36.3·15-s + 16·16-s − 66.5·17-s + 18·18-s + 111.·19-s + 48.5·20-s + 21·21-s + 20.2·22-s + 23·23-s + 24·24-s + 22.2·25-s − 5.56·26-s + 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.08·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.767·10-s + 0.277·11-s + 0.288·12-s − 0.0593·13-s + 0.267·14-s + 0.626·15-s + 0.250·16-s − 0.950·17-s + 0.235·18-s + 1.34·19-s + 0.542·20-s + 0.218·21-s + 0.196·22-s + 0.208·23-s + 0.204·24-s + 0.177·25-s − 0.0419·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.702255637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.702255637\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 - 12.1T + 125T^{2} \) |
| 11 | \( 1 - 10.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.78T + 2.19e3T^{2} \) |
| 17 | \( 1 + 66.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 111.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 52.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 223.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 212.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 42.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 351.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 416.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 640.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 42.6T + 2.05e5T^{2} \) |
| 61 | \( 1 - 7.06T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.04e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 185.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 202.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 336.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 681.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 264.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.19e3T + 9.12e5T^{2} \) |
| 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
−9.710083226442479684529274352256, −8.856596188598615637068800369060, −7.931083893157842814099582429492, −6.92648084975644773728612092951, −6.16683566086742425311865330150, −5.19923900342414425582861781692, −4.39508399340713750163660585766, −3.16433005154586950767634701360, −2.26487733769098236735197582854, −1.26501823073020756578687736316,
1.26501823073020756578687736316, 2.26487733769098236735197582854, 3.16433005154586950767634701360, 4.39508399340713750163660585766, 5.19923900342414425582861781692, 6.16683566086742425311865330150, 6.92648084975644773728612092951, 7.931083893157842814099582429492, 8.856596188598615637068800369060, 9.710083226442479684529274352256
