| L(s) = 1 | − 2-s + 4-s + 1.65·5-s + 0.0358·7-s − 8-s − 1.65·10-s − 1.10·11-s + 1.20·13-s − 0.0358·14-s + 16-s + 3.50·17-s + 4.17·19-s + 1.65·20-s + 1.10·22-s + 5.77·23-s − 2.26·25-s − 1.20·26-s + 0.0358·28-s − 8.50·29-s − 7.39·31-s − 32-s − 3.50·34-s + 0.0592·35-s − 8.83·37-s − 4.17·38-s − 1.65·40-s + 5.77·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.739·5-s + 0.0135·7-s − 0.353·8-s − 0.522·10-s − 0.334·11-s + 0.333·13-s − 0.00958·14-s + 0.250·16-s + 0.849·17-s + 0.958·19-s + 0.369·20-s + 0.236·22-s + 1.20·23-s − 0.453·25-s − 0.235·26-s + 0.00677·28-s − 1.57·29-s − 1.32·31-s − 0.176·32-s − 0.600·34-s + 0.0100·35-s − 1.45·37-s − 0.677·38-s − 0.261·40-s + 0.901·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 - 1.65T + 5T^{2} \) |
| 7 | \( 1 - 0.0358T + 7T^{2} \) |
| 11 | \( 1 + 1.10T + 11T^{2} \) |
| 13 | \( 1 - 1.20T + 13T^{2} \) |
| 17 | \( 1 - 3.50T + 17T^{2} \) |
| 19 | \( 1 - 4.17T + 19T^{2} \) |
| 23 | \( 1 - 5.77T + 23T^{2} \) |
| 29 | \( 1 + 8.50T + 29T^{2} \) |
| 31 | \( 1 + 7.39T + 31T^{2} \) |
| 37 | \( 1 + 8.83T + 37T^{2} \) |
| 41 | \( 1 - 5.77T + 41T^{2} \) |
| 43 | \( 1 + 7.59T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + 8.49T + 53T^{2} \) |
| 59 | \( 1 + 9.24T + 59T^{2} \) |
| 61 | \( 1 + 6.48T + 61T^{2} \) |
| 67 | \( 1 + 3.00T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 3.64T + 73T^{2} \) |
| 79 | \( 1 - 4.74T + 79T^{2} \) |
| 83 | \( 1 + 2.78T + 83T^{2} \) |
| 89 | \( 1 - 6.07T + 89T^{2} \) |
| 97 | \( 1 - 0.383T + 97T^{2} \) |
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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
−7.64910666768568380356855865942, −6.88428752011937240470284516681, −6.16036478034921990391203786195, −5.40813258154699120206792042342, −4.99865863412915804372792554881, −3.53487679332636767639253806371, −3.14566934229000785154846051210, −1.90155157901033363859471150659, −1.39961464549881625754258589737, 0,
1.39961464549881625754258589737, 1.90155157901033363859471150659, 3.14566934229000785154846051210, 3.53487679332636767639253806371, 4.99865863412915804372792554881, 5.40813258154699120206792042342, 6.16036478034921990391203786195, 6.88428752011937240470284516681, 7.64910666768568380356855865942
