L(s) = 1 | − 5-s − 2·7-s + 3·11-s − 13-s + 4·23-s − 4·25-s + 29-s − 3·31-s + 2·35-s − 8·37-s + 6·41-s + 5·43-s + 3·47-s − 3·49-s − 5·53-s − 3·55-s − 8·59-s + 65-s + 12·67-s + 6·71-s − 4·73-s − 6·77-s − 79-s − 12·83-s − 6·89-s + 2·91-s + 14·97-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.755·7-s + 0.904·11-s − 0.277·13-s + 0.834·23-s − 4/5·25-s + 0.185·29-s − 0.538·31-s + 0.338·35-s − 1.31·37-s + 0.937·41-s + 0.762·43-s + 0.437·47-s − 3/7·49-s − 0.686·53-s − 0.404·55-s − 1.04·59-s + 0.124·65-s + 1.46·67-s + 0.712·71-s − 0.468·73-s − 0.683·77-s − 0.112·79-s − 1.31·83-s − 0.635·89-s + 0.209·91-s + 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 29 | \( 1 - T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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
−7.994610179369606995869884378763, −7.24961226348188201527304882529, −6.65320811852602296876789762362, −5.92169113770517864413828542879, −5.03803556130782721048021606314, −4.08187314982047767565464280256, −3.50289320210776225141646880508, −2.56353613863743857251285305435, −1.33307717435361558517907963551, 0,
1.33307717435361558517907963551, 2.56353613863743857251285305435, 3.50289320210776225141646880508, 4.08187314982047767565464280256, 5.03803556130782721048021606314, 5.92169113770517864413828542879, 6.65320811852602296876789762362, 7.24961226348188201527304882529, 7.994610179369606995869884378763