| L(s) = 1 | − i·3-s + (2 − i)5-s − 4i·7-s − 9-s − 6i·13-s + (−1 − 2i)15-s − 2i·17-s − 6·19-s − 4·21-s + 6i·23-s + (3 − 4i)25-s + i·27-s + 8·29-s − 8·31-s + (−4 − 8i)35-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (0.894 − 0.447i)5-s − 1.51i·7-s − 0.333·9-s − 1.66i·13-s + (−0.258 − 0.516i)15-s − 0.485i·17-s − 1.37·19-s − 0.872·21-s + 1.25i·23-s + (0.600 − 0.800i)25-s + 0.192i·27-s + 1.48·29-s − 1.43·31-s + (−0.676 − 1.35i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.655820207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.655820207\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-2 + i)T \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.677837639968309451837005472450, −8.121732002480809353675107529992, −7.21516605585194194959190067794, −6.62294726370191494405706318762, −5.63860816953378232698023246273, −4.95065225135551634082598349079, −3.83291737925194375357772197923, −2.79139402660676206332875153889, −1.52897513463252493945054421534, −0.58184425971407767233654672679,
2.05373433774555767093572023145, 2.39645388668815366288858194558, 3.76119844539937844423297141054, 4.73602281260995820595505859434, 5.57182273236609865220494082500, 6.37911386646073402901520764011, 6.79989865345280484695001224396, 8.392865466590526959443821282965, 8.852788276641653995601695500925, 9.406171627091146240017670032624
