L(s) = 1 | − 2-s + 4-s + 5-s − 2·7-s − 8-s − 10-s + 2·11-s − 6·13-s + 2·14-s + 16-s + 4·17-s + 20-s − 2·22-s − 23-s + 25-s + 6·26-s − 2·28-s − 2·29-s − 32-s − 4·34-s − 2·35-s − 8·37-s − 40-s + 6·41-s − 4·43-s + 2·44-s + 46-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.755·7-s − 0.353·8-s − 0.316·10-s + 0.603·11-s − 1.66·13-s + 0.534·14-s + 1/4·16-s + 0.970·17-s + 0.223·20-s − 0.426·22-s − 0.208·23-s + 1/5·25-s + 1.17·26-s − 0.377·28-s − 0.371·29-s − 0.176·32-s − 0.685·34-s − 0.338·35-s − 1.31·37-s − 0.158·40-s + 0.937·41-s − 0.609·43-s + 0.301·44-s + 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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
−8.973278703184901854138048170284, −7.895261877145436396222250083702, −7.24355455933227649972572881776, −6.51306565684234503632733201847, −5.70466704477790752725776214885, −4.78496393552877477867010504333, −3.50821920572786566885575819128, −2.63682075596531236017404810466, −1.53167294603489718963966945293, 0,
1.53167294603489718963966945293, 2.63682075596531236017404810466, 3.50821920572786566885575819128, 4.78496393552877477867010504333, 5.70466704477790752725776214885, 6.51306565684234503632733201847, 7.24355455933227649972572881776, 7.895261877145436396222250083702, 8.973278703184901854138048170284