| L(s) = 1 | + 2-s + 3-s + 4-s − 2·5-s + 6-s + 7-s + 8-s − 2·9-s − 2·10-s − 11-s + 12-s + 14-s − 2·15-s + 16-s − 3·17-s − 2·18-s + 6·19-s − 2·20-s + 21-s − 22-s + 6·23-s + 24-s − 25-s − 5·27-s + 28-s − 4·29-s − 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.632·10-s − 0.301·11-s + 0.288·12-s + 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.727·17-s − 0.471·18-s + 1.37·19-s − 0.447·20-s + 0.218·21-s − 0.213·22-s + 1.25·23-s + 0.204·24-s − 1/5·25-s − 0.962·27-s + 0.188·28-s − 0.742·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−15.38776814939069, −14.94989183742362, −14.72258441731773, −13.83591216257438, −13.46384856117115, −13.25020547219166, −12.09342861723482, −11.97231682944220, −11.35998123792081, −10.97962664402850, −10.24011165240498, −9.513664982983915, −8.883250411161663, −8.276938959650409, −7.902636794946852, −7.212890549620104, −6.785495495573941, −5.868123118457926, −5.240959122030457, −4.768194825867526, −4.002770766637162, −3.346131190370764, −2.919432509189847, −2.153558721900293, −1.174920521959985, 0,
1.174920521959985, 2.153558721900293, 2.919432509189847, 3.346131190370764, 4.002770766637162, 4.768194825867526, 5.240959122030457, 5.868123118457926, 6.785495495573941, 7.212890549620104, 7.902636794946852, 8.276938959650409, 8.883250411161663, 9.513664982983915, 10.24011165240498, 10.97962664402850, 11.35998123792081, 11.97231682944220, 12.09342861723482, 13.25020547219166, 13.46384856117115, 13.83591216257438, 14.72258441731773, 14.94989183742362, 15.38776814939069
