| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 5·5-s − 6·6-s − 32·7-s − 8·8-s + 9·9-s + 10·10-s + 27·11-s + 12·12-s − 69·13-s + 64·14-s − 15·15-s + 16·16-s − 17·17-s − 18·18-s − 83·19-s − 20·20-s − 96·21-s − 54·22-s − 117·23-s − 24·24-s − 100·25-s + 138·26-s + 27·27-s − 128·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.72·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.740·11-s + 0.288·12-s − 1.47·13-s + 1.22·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 1.00·19-s − 0.223·20-s − 0.997·21-s − 0.523·22-s − 1.06·23-s − 0.204·24-s − 4/5·25-s + 1.04·26-s + 0.192·27-s − 0.863·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 3 | \( 1 - p T \) |
| 17 | \( 1 + p T \) |
| good | 5 | \( 1 + p T + p^{3} T^{2} \) |
| 7 | \( 1 + 32 T + p^{3} T^{2} \) |
| 11 | \( 1 - 27 T + p^{3} T^{2} \) |
| 13 | \( 1 + 69 T + p^{3} T^{2} \) |
| 19 | \( 1 + 83 T + p^{3} T^{2} \) |
| 23 | \( 1 + 117 T + p^{3} T^{2} \) |
| 29 | \( 1 - 94 T + p^{3} T^{2} \) |
| 31 | \( 1 - 198 T + p^{3} T^{2} \) |
| 37 | \( 1 + 244 T + p^{3} T^{2} \) |
| 41 | \( 1 - 169 T + p^{3} T^{2} \) |
| 43 | \( 1 - 227 T + p^{3} T^{2} \) |
| 47 | \( 1 + 382 T + p^{3} T^{2} \) |
| 53 | \( 1 - 686 T + p^{3} T^{2} \) |
| 59 | \( 1 - 450 T + p^{3} T^{2} \) |
| 61 | \( 1 + 700 T + p^{3} T^{2} \) |
| 67 | \( 1 - 540 T + p^{3} T^{2} \) |
| 71 | \( 1 + 276 T + p^{3} T^{2} \) |
| 73 | \( 1 + 298 T + p^{3} T^{2} \) |
| 79 | \( 1 + 182 T + p^{3} T^{2} \) |
| 83 | \( 1 - 282 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1468 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1140 T + p^{3} T^{2} \) |
| 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
−12.62263491761558122646019510463, −11.91837967874973408845072587698, −10.20010758645313232167557824297, −9.623955939677458321012019750180, −8.532718634208962779844451596608, −7.24360626005749331927364216728, −6.31491209981510445875409246915, −3.99778165279163441863197410079, −2.55322158194758546896632343121, 0,
2.55322158194758546896632343121, 3.99778165279163441863197410079, 6.31491209981510445875409246915, 7.24360626005749331927364216728, 8.532718634208962779844451596608, 9.623955939677458321012019750180, 10.20010758645313232167557824297, 11.91837967874973408845072587698, 12.62263491761558122646019510463
