| L(s) = 1 | − 2·2-s − 7.18·3-s + 4·4-s + 20.3·5-s + 14.3·6-s − 8·8-s + 24.5·9-s − 40.6·10-s + 1.47·11-s − 28.7·12-s − 32.4·13-s − 146.·15-s + 16·16-s − 31.8·17-s − 49.1·18-s + 51.2·19-s + 81.3·20-s − 2.94·22-s + 23·23-s + 57.4·24-s + 288.·25-s + 64.9·26-s + 17.2·27-s − 39.6·29-s + 292.·30-s − 35.8·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.38·3-s + 0.5·4-s + 1.81·5-s + 0.977·6-s − 0.353·8-s + 0.910·9-s − 1.28·10-s + 0.0404·11-s − 0.691·12-s − 0.693·13-s − 2.51·15-s + 0.250·16-s − 0.454·17-s − 0.644·18-s + 0.619·19-s + 0.909·20-s − 0.0285·22-s + 0.208·23-s + 0.488·24-s + 2.30·25-s + 0.490·26-s + 0.123·27-s − 0.254·29-s + 1.77·30-s − 0.207·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 2T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 7.18T + 27T^{2} \) |
| 5 | \( 1 - 20.3T + 125T^{2} \) |
| 11 | \( 1 - 1.47T + 1.33e3T^{2} \) |
| 13 | \( 1 + 32.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 31.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 51.2T + 6.85e3T^{2} \) |
| 29 | \( 1 + 39.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 35.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 76.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 162.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 336.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 215.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 304.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 97.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 923.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 604.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 769.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 400.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 423.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 437.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 651.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 610.T + 9.12e5T^{2} \) |
| 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
−8.456755783156349579338634414598, −7.23875880482781577991161321474, −6.62473007121011835757683212925, −5.96920412755300035481177332274, −5.36580245042089016495516414085, −4.70446511110882892327510367785, −2.97557446186758137968100033283, −1.96831207865695555260835850335, −1.14776719376126404069832093661, 0,
1.14776719376126404069832093661, 1.96831207865695555260835850335, 2.97557446186758137968100033283, 4.70446511110882892327510367785, 5.36580245042089016495516414085, 5.96920412755300035481177332274, 6.62473007121011835757683212925, 7.23875880482781577991161321474, 8.456755783156349579338634414598
