| L(s) = 1 | + 0.709·2-s + 3·3-s − 7.49·4-s + 5.73·5-s + 2.12·6-s + 14.2·7-s − 10.9·8-s + 9·9-s + 4.06·10-s − 11·11-s − 22.4·12-s + 39.9·13-s + 10.1·14-s + 17.2·15-s + 52.1·16-s − 105.·17-s + 6.38·18-s − 127.·19-s − 42.9·20-s + 42.8·21-s − 7.80·22-s + 89.5·23-s − 32.9·24-s − 92.1·25-s + 28.3·26-s + 27·27-s − 107.·28-s + ⋯ |
| L(s) = 1 | + 0.250·2-s + 0.577·3-s − 0.937·4-s + 0.512·5-s + 0.144·6-s + 0.770·7-s − 0.485·8-s + 0.333·9-s + 0.128·10-s − 0.301·11-s − 0.541·12-s + 0.851·13-s + 0.193·14-s + 0.296·15-s + 0.815·16-s − 1.50·17-s + 0.0835·18-s − 1.53·19-s − 0.480·20-s + 0.445·21-s − 0.0755·22-s + 0.811·23-s − 0.280·24-s − 0.736·25-s + 0.213·26-s + 0.192·27-s − 0.722·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 - 0.709T + 8T^{2} \) |
| 5 | \( 1 - 5.73T + 125T^{2} \) |
| 7 | \( 1 - 14.2T + 343T^{2} \) |
| 13 | \( 1 - 39.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 105.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 89.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 12.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 229.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 196.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 452.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 177.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 338.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 687.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 428.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 529.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 993.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 113.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 173.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 687.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 25.8T + 7.04e5T^{2} \) |
| 97 | \( 1 - 216.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.306085072812602775257959198171, −8.106074577128828735552270835205, −6.63951229113180558328716830774, −6.06997292013286491965419374226, −4.81639784430640052255999378009, −4.51465517376080637216704084080, −3.47168346621393221701221424216, −2.36566130625483465847930370069, −1.41886449402990808699998908862, 0,
1.41886449402990808699998908862, 2.36566130625483465847930370069, 3.47168346621393221701221424216, 4.51465517376080637216704084080, 4.81639784430640052255999378009, 6.06997292013286491965419374226, 6.63951229113180558328716830774, 8.106074577128828735552270835205, 8.306085072812602775257959198171
