L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 2·7-s − 8-s + 9-s + 2·11-s − 12-s − 4·13-s − 2·14-s + 16-s + 2·17-s − 18-s − 19-s − 2·21-s − 2·22-s − 4·23-s + 24-s + 4·26-s − 27-s + 2·28-s − 8·31-s − 32-s − 2·33-s − 2·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.288·12-s − 1.10·13-s − 0.534·14-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.229·19-s − 0.436·21-s − 0.426·22-s − 0.834·23-s + 0.204·24-s + 0.784·26-s − 0.192·27-s + 0.377·28-s − 1.43·31-s − 0.176·32-s − 0.348·33-s − 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−8.522562468973123460133829683234, −7.38034517609703092261621472438, −7.28883706759817309640759737976, −6.10918521028023032139338446081, −5.42826853188311424543299005218, −4.57330627362562972969370314906, −3.60546983722633947720313363909, −2.25825312991053357304272575859, −1.41053369409238720666338505426, 0,
1.41053369409238720666338505426, 2.25825312991053357304272575859, 3.60546983722633947720313363909, 4.57330627362562972969370314906, 5.42826853188311424543299005218, 6.10918521028023032139338446081, 7.28883706759817309640759737976, 7.38034517609703092261621472438, 8.522562468973123460133829683234