| L(s) = 1 | + i·7-s + 4·11-s − i·13-s + 4i·17-s + 19-s − 4i·23-s + 4·29-s + 5·31-s + 6i·37-s − 12·41-s − 5i·43-s − 8i·47-s + 6·49-s + 12i·53-s − 8·59-s + ⋯ |
| L(s) = 1 | + 0.377i·7-s + 1.20·11-s − 0.277i·13-s + 0.970i·17-s + 0.229·19-s − 0.834i·23-s + 0.742·29-s + 0.898·31-s + 0.986i·37-s − 1.87·41-s − 0.762i·43-s − 1.16i·47-s + 0.857·49-s + 1.64i·53-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.043869196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.043869196\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 + 5iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 - 13iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 13iT - 97T^{2} \) |
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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
−8.559329833718780737585645084983, −8.091134967259805296762080742884, −6.90930272337620533188762753234, −6.47840349335689484587310372512, −5.67605902909172650935027472209, −4.76283793191440230393048767946, −3.96169618313518412497596125228, −3.11997496295862470772801122941, −2.04605069918607462848912625190, −0.978429553220906821166989932321,
0.78241731799460590624972547120, 1.82632277824063517047850333270, 3.04505998799951721824141860619, 3.83167034023701742621503005026, 4.65734777800103942978237311443, 5.42225340581984716543995904422, 6.51902654231889450183513660349, 6.85126321159588902687012436004, 7.76455836642957990418026054521, 8.475228181135524051090930607721
