| L(s) = 1 | − 3-s + 4i·5-s − i·7-s + 9-s + (−3 + 2i)13-s − 4i·15-s − 2·17-s + i·21-s − 4·23-s − 11·25-s − 27-s + 2·29-s − 8i·31-s + 4·35-s − 8i·37-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78i·5-s − 0.377i·7-s + 0.333·9-s + (−0.832 + 0.554i)13-s − 1.03i·15-s − 0.485·17-s + 0.218i·21-s − 0.834·23-s − 2.20·25-s − 0.192·27-s + 0.371·29-s − 1.43i·31-s + 0.676·35-s − 1.31i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6317958989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6317958989\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (3 - 2i)T \) |
| good | 5 | \( 1 - 4iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 8iT - 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 4iT - 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 8iT - 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 - 12iT - 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−7.87197703827494062877431580256, −7.42341071967536358943451076175, −6.76674053423287759388225383093, −6.22737762950374223145311289812, −5.48477662326818292399031119671, −4.30040158815505430621987789687, −3.79749672652095746115215417227, −2.62953165904748964823325901800, −2.03895980514155607256953307172, −0.22735640442195413357593259154,
0.913217991921513170717520337747, 1.84329804262241222572718503483, 3.04052540637661307078577740902, 4.44897987779151545281859609022, 4.63809081252335917557250494370, 5.53879240361378640278284856046, 5.98872454516788495442732501366, 7.06951750335072134300135227031, 7.86704772750699720547427361350, 8.592648872227311680537321122054
