L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 4·5-s − 2·6-s − 2·7-s + 9-s + 8·10-s − 3·11-s + 2·12-s − 6·13-s + 4·14-s − 4·15-s − 4·16-s + 3·17-s − 2·18-s − 8·20-s − 2·21-s + 6·22-s − 6·23-s + 11·25-s + 12·26-s + 27-s − 4·28-s + 5·29-s + 8·30-s + 7·31-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 1.78·5-s − 0.816·6-s − 0.755·7-s + 1/3·9-s + 2.52·10-s − 0.904·11-s + 0.577·12-s − 1.66·13-s + 1.06·14-s − 1.03·15-s − 16-s + 0.727·17-s − 0.471·18-s − 1.78·20-s − 0.436·21-s + 1.27·22-s − 1.25·23-s + 11/5·25-s + 2.35·26-s + 0.192·27-s − 0.755·28-s + 0.928·29-s + 1.46·30-s + 1.25·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 41 | \( 1 - T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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
−12.55068102055940082127041478018, −11.83698209569361769057259938325, −10.40776482924956216013302313541, −9.765611747485775303186116359740, −8.409047961877697867726744185287, −7.80091344637969470324860470762, −7.04855950160821631603912974995, −4.53745508909086524538385836361, −2.91545403229970325971647429959, 0,
2.91545403229970325971647429959, 4.53745508909086524538385836361, 7.04855950160821631603912974995, 7.80091344637969470324860470762, 8.409047961877697867726744185287, 9.765611747485775303186116359740, 10.40776482924956216013302313541, 11.83698209569361769057259938325, 12.55068102055940082127041478018