| L(s) = 1 | + 2-s − 3.30·3-s + 4-s + 5-s − 3.30·6-s − 4.60·7-s + 8-s + 7.90·9-s + 10-s − 11-s − 3.30·12-s − 0.697·13-s − 4.60·14-s − 3.30·15-s + 16-s + 7.90·18-s + 4.30·19-s + 20-s + 15.2·21-s − 22-s − 2·23-s − 3.30·24-s + 25-s − 0.697·26-s − 16.2·27-s − 4.60·28-s + 0.605·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.90·3-s + 0.5·4-s + 0.447·5-s − 1.34·6-s − 1.74·7-s + 0.353·8-s + 2.63·9-s + 0.316·10-s − 0.301·11-s − 0.953·12-s − 0.193·13-s − 1.23·14-s − 0.852·15-s + 0.250·16-s + 1.86·18-s + 0.987·19-s + 0.223·20-s + 3.31·21-s − 0.213·22-s − 0.417·23-s − 0.674·24-s + 0.200·25-s − 0.136·26-s − 3.11·27-s − 0.870·28-s + 0.112·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 3 | \( 1 + 3.30T + 3T^{2} \) |
| 7 | \( 1 + 4.60T + 7T^{2} \) |
| 13 | \( 1 + 0.697T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 4.30T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 - 0.605T + 29T^{2} \) |
| 31 | \( 1 + 3.39T + 31T^{2} \) |
| 37 | \( 1 + 7.21T + 37T^{2} \) |
| 41 | \( 1 - 7.21T + 41T^{2} \) |
| 43 | \( 1 - 5.51T + 43T^{2} \) |
| 47 | \( 1 + 4.30T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 2.09T + 59T^{2} \) |
| 61 | \( 1 - 0.697T + 61T^{2} \) |
| 67 | \( 1 - 3.30T + 67T^{2} \) |
| 71 | \( 1 - 4.30T + 71T^{2} \) |
| 79 | \( 1 - 9.81T + 79T^{2} \) |
| 83 | \( 1 - 6.51T + 83T^{2} \) |
| 89 | \( 1 + 7.30T + 89T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
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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
−7.02434539017719998594585622776, −6.60477078693055558519950737023, −5.95708230836506924135633791544, −5.54124184765967364898838698502, −4.94350608543307355188239875026, −4.01667866470199250735383621312, −3.32843137497293816614415690569, −2.25753407754343833202578757484, −1.02451740253072986691224536623, 0,
1.02451740253072986691224536623, 2.25753407754343833202578757484, 3.32843137497293816614415690569, 4.01667866470199250735383621312, 4.94350608543307355188239875026, 5.54124184765967364898838698502, 5.95708230836506924135633791544, 6.60477078693055558519950737023, 7.02434539017719998594585622776
