| L(s) = 1 | + 3.61·2-s + 10.0·3-s + 5.03·4-s + 15.3·5-s + 36.3·6-s − 7·7-s − 10.7·8-s + 74.5·9-s + 55.2·10-s + 5.17·11-s + 50.7·12-s + 32.5·13-s − 25.2·14-s + 154.·15-s − 78.9·16-s + 269.·18-s − 98.4·19-s + 77.1·20-s − 70.5·21-s + 18.6·22-s + 185.·23-s − 107.·24-s + 109.·25-s + 117.·26-s + 479.·27-s − 35.2·28-s + 43.6·29-s + ⋯ |
| L(s) = 1 | + 1.27·2-s + 1.93·3-s + 0.629·4-s + 1.36·5-s + 2.47·6-s − 0.377·7-s − 0.472·8-s + 2.76·9-s + 1.74·10-s + 0.141·11-s + 1.22·12-s + 0.694·13-s − 0.482·14-s + 2.65·15-s − 1.23·16-s + 3.52·18-s − 1.18·19-s + 0.862·20-s − 0.733·21-s + 0.180·22-s + 1.68·23-s − 0.917·24-s + 0.875·25-s + 0.886·26-s + 3.41·27-s − 0.237·28-s + 0.279·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(11.96459977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.96459977\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 7T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 3.61T + 8T^{2} \) |
| 3 | \( 1 - 10.0T + 27T^{2} \) |
| 5 | \( 1 - 15.3T + 125T^{2} \) |
| 11 | \( 1 - 5.17T + 1.33e3T^{2} \) |
| 13 | \( 1 - 32.5T + 2.19e3T^{2} \) |
| 19 | \( 1 + 98.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 185.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 43.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 80.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 297.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 429.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 311.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 261.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 346.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 151.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 451.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 628.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 110.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 899.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 68.7T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.41e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.39e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 625.T + 9.12e5T^{2} \) |
| 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
−8.927509433943140401159685085599, −8.230090156328174213793024355886, −6.88188395360541619229479045462, −6.52328092138877645225450155060, −5.45114797598591437341597401191, −4.52010025839443047432985688889, −3.72067981956377904021691695495, −2.93800285011168652796558664516, −2.34379084186886096243054094215, −1.37849847007733955729076470146,
1.37849847007733955729076470146, 2.34379084186886096243054094215, 2.93800285011168652796558664516, 3.72067981956377904021691695495, 4.52010025839443047432985688889, 5.45114797598591437341597401191, 6.52328092138877645225450155060, 6.88188395360541619229479045462, 8.230090156328174213793024355886, 8.927509433943140401159685085599
