| L(s) = 1 | + i·3-s − 9-s + 4·11-s − 6i·13-s − 6i·17-s − 4·19-s − i·27-s + 2·29-s + 8·31-s + 4i·33-s − 2i·37-s + 6·39-s − 6·41-s + 12i·43-s − 8i·47-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.333·9-s + 1.20·11-s − 1.66i·13-s − 1.45i·17-s − 0.917·19-s − 0.192i·27-s + 0.371·29-s + 1.43·31-s + 0.696i·33-s − 0.328i·37-s + 0.960·39-s − 0.937·41-s + 1.82i·43-s − 1.16i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.605797357\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.605797357\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 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 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−9.931262264743306093004710089817, −8.814598430926268027923828107991, −8.277529208404229630937827649933, −7.16580120297517541933710892352, −6.32015188731403822706283273449, −5.35593063783492880148621853191, −4.52770601332007010236933037983, −3.50847780841023239525392880373, −2.56673412766581575731121058677, −0.77146599707915172535366148985,
1.34273256913335249576489420451, 2.27243183320544802701974579879, 3.83277241589066976058645379695, 4.42442984847417417168818826386, 5.87769062104612673901513864072, 6.59658709041959294152260972788, 7.07036372701954498718765370591, 8.489067701080944716754451533930, 8.683498348650710752501376736498, 9.773779295038427916214114622401
