| L(s) = 1 | − 2-s + 8·3-s − 7·4-s + 5·5-s − 8·6-s + 14·7-s + 15·8-s + 37·9-s − 5·10-s − 20·11-s − 56·12-s − 58·13-s − 14·14-s + 40·15-s + 41·16-s − 37·18-s − 80·19-s − 35·20-s + 112·21-s + 20·22-s − 118·23-s + 120·24-s + 25·25-s + 58·26-s + 80·27-s − 98·28-s + 126·29-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 1.53·3-s − 7/8·4-s + 0.447·5-s − 0.544·6-s + 0.755·7-s + 0.662·8-s + 1.37·9-s − 0.158·10-s − 0.548·11-s − 1.34·12-s − 1.23·13-s − 0.267·14-s + 0.688·15-s + 0.640·16-s − 0.484·18-s − 0.965·19-s − 0.391·20-s + 1.16·21-s + 0.193·22-s − 1.06·23-s + 1.02·24-s + 1/5·25-s + 0.437·26-s + 0.570·27-s − 0.661·28-s + 0.806·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - p T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 3 | \( 1 - 8 T + p^{3} T^{2} \) |
| 7 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 20 T + p^{3} T^{2} \) |
| 13 | \( 1 + 58 T + p^{3} T^{2} \) |
| 19 | \( 1 + 80 T + p^{3} T^{2} \) |
| 23 | \( 1 + 118 T + p^{3} T^{2} \) |
| 29 | \( 1 - 126 T + p^{3} T^{2} \) |
| 31 | \( 1 - 70 T + p^{3} T^{2} \) |
| 37 | \( 1 + 134 T + p^{3} T^{2} \) |
| 41 | \( 1 - 100 T + p^{3} T^{2} \) |
| 43 | \( 1 + 272 T + p^{3} T^{2} \) |
| 47 | \( 1 + 464 T + p^{3} T^{2} \) |
| 53 | \( 1 + 642 T + p^{3} T^{2} \) |
| 59 | \( 1 - 180 T + p^{3} T^{2} \) |
| 61 | \( 1 + 110 T + p^{3} T^{2} \) |
| 67 | \( 1 + 924 T + p^{3} T^{2} \) |
| 71 | \( 1 - 90 T + p^{3} T^{2} \) |
| 73 | \( 1 - 828 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1334 T + p^{3} T^{2} \) |
| 83 | \( 1 + 552 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1490 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1376 T + p^{3} T^{2} \) |
| 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
−8.694810670537401043781598358947, −8.009095523395774108725463933466, −7.75147069093604048490937899505, −6.46612894432150299717748344443, −5.04168764613039770417784714345, −4.56337974470615541296332637086, −3.44735092285437451257547019786, −2.37674921964533963873605159019, −1.63424596339482039275841213067, 0,
1.63424596339482039275841213067, 2.37674921964533963873605159019, 3.44735092285437451257547019786, 4.56337974470615541296332637086, 5.04168764613039770417784714345, 6.46612894432150299717748344443, 7.75147069093604048490937899505, 8.009095523395774108725463933466, 8.694810670537401043781598358947
