L(s) = 1 | + i·7-s + 3·9-s + 4·11-s − 2i·13-s − 6i·17-s + 8·19-s − 6·29-s − 8·31-s − 2i·37-s + 2·41-s − 4i·43-s + 8i·47-s − 49-s − 6i·53-s − 6·61-s + ⋯ |
L(s) = 1 | + 0.377i·7-s + 9-s + 1.20·11-s − 0.554i·13-s − 1.45i·17-s + 1.83·19-s − 1.11·29-s − 1.43·31-s − 0.328i·37-s + 0.312·41-s − 0.609i·43-s + 1.16i·47-s − 0.142·49-s − 0.824i·53-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.212451677\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.212451677\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 - 3T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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
−9.136593376641330279522091652207, −7.69132662829273785148995926677, −7.38068957337938074473716968313, −6.56078834858790858057240217710, −5.54620784808665704746967987922, −4.95973632499220362464177764208, −3.87228518123592236766305932970, −3.17994126280343623931590566976, −1.91157316119000157145512689155, −0.856631179434997172164991284720,
1.18792464571850788663944682684, 1.87822380355054306791066199671, 3.58354730656765757757029358982, 3.87933605204572010252609008312, 4.89344016659079546967315912208, 5.86569624183003536585035667707, 6.67654802923301802711668886673, 7.32067312617573388412029045281, 7.948249173357360129596475598299, 9.132286237272849379631871086040