| 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 | \( 1 + iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.265269534654954141410990237911, −7.53552221236446321492870944747, −7.33765788628101262310077533277, −6.48103681477865901897981336596, −5.57922927235198182161478566706, −4.95080524062542653579973794126, −3.74406004926410892658320722383, −3.38409052496117103209751156006, −2.27551006542102180849151253786, −1.16649143083525911993073084617,
0.21861776551954550649651597637, 1.80757890890138271899321401355, 2.47766757880110020421314770561, 3.23825232365821563417356673251, 4.74262201340020394324254756137, 5.19159696553250854489100532840, 5.63204685621329517791850004295, 6.45410273965163771018212917258, 7.47015492295014851257914693387, 8.401311192729721344580035422274
