| L(s) = 1 | + (−2 + i)5-s − 2i·7-s + 2·11-s − 6i·13-s + i·17-s + 3·19-s − 7i·23-s + (3 − 4i)25-s + 6·29-s + 31-s + (2 + 4i)35-s − 8i·37-s − 10·41-s + 2i·43-s + 8i·47-s + ⋯ |
| L(s) = 1 | + (−0.894 + 0.447i)5-s − 0.755i·7-s + 0.603·11-s − 1.66i·13-s + 0.242i·17-s + 0.688·19-s − 1.45i·23-s + (0.600 − 0.800i)25-s + 1.11·29-s + 0.179·31-s + (0.338 + 0.676i)35-s − 1.31i·37-s − 1.56·41-s + 0.304i·43-s + 1.16i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.974356 - 0.602185i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.974356 - 0.602185i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 - i)T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - iT - 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + 7iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 15T + 79T^{2} \) |
| 83 | \( 1 + 13iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
| 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
−10.54939732165623187725831600614, −10.12912540563452955944081984675, −8.685923469006776668757095878589, −7.932628517368710060976294001566, −7.14212536377970532250190407346, −6.21835325972148844419345086621, −4.86582177645926965376547087074, −3.82256187758409686623163705284, −2.89484195594571694937824998390, −0.73024365488627143114837307951,
1.55404405174510004775783383457, 3.23736979727494902191180070795, 4.33121207109617791024599879894, 5.24997908052656329581298993067, 6.53018612404254597605459097087, 7.35246687627331741518944103509, 8.485715944527938870052490067798, 9.079301038216793549547981869143, 9.907555653910551580120790769048, 11.38515460673658854886615464507
