| L(s) = 1 | + (1 − 2i)5-s + 5i·7-s − 5·11-s + i·13-s − 3i·17-s + 4·19-s − 5i·23-s + (−3 − 4i)25-s − 4·29-s + (10 + 5i)35-s + 7i·37-s − 11·41-s − 12i·43-s + 6i·47-s − 18·49-s + ⋯ |
| L(s) = 1 | + (0.447 − 0.894i)5-s + 1.88i·7-s − 1.50·11-s + 0.277i·13-s − 0.727i·17-s + 0.917·19-s − 1.04i·23-s + (−0.600 − 0.800i)25-s − 0.742·29-s + (1.69 + 0.845i)35-s + 1.15i·37-s − 1.71·41-s − 1.82i·43-s + 0.875i·47-s − 2.57·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8068359490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8068359490\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1 + 2i)T \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 - 5iT - 7T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 5iT - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 7iT - 37T^{2} \) |
| 41 | \( 1 + 11T + 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 7T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 - 2iT - 83T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 - iT - 97T^{2} \) |
| 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
−8.401311192729721344580035422274, −7.47015492295014851257914693387, −6.45410273965163771018212917258, −5.63204685621329517791850004295, −5.19159696553250854489100532840, −4.74262201340020394324254756137, −3.23825232365821563417356673251, −2.47766757880110020421314770561, −1.80757890890138271899321401355, −0.21861776551954550649651597637,
1.16649143083525911993073084617, 2.27551006542102180849151253786, 3.38409052496117103209751156006, 3.74406004926410892658320722383, 4.95080524062542653579973794126, 5.57922927235198182161478566706, 6.48103681477865901897981336596, 7.33765788628101262310077533277, 7.53552221236446321492870944747, 8.265269534654954141410990237911
