| L(s) = 1 | + i·3-s + i·7-s − 9-s − 6i·13-s − 2i·17-s − 4·19-s − 21-s + 4i·23-s − i·27-s + 10·29-s − 8·31-s + 6i·37-s + 6·39-s − 2·41-s + 4i·43-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.377i·7-s − 0.333·9-s − 1.66i·13-s − 0.485i·17-s − 0.917·19-s − 0.218·21-s + 0.834i·23-s − 0.192i·27-s + 1.85·29-s − 1.43·31-s + 0.986i·37-s + 0.960·39-s − 0.312·41-s + 0.609i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7578276384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7578276384\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.718984736885627561507816816872, −8.059603729085936049659807560541, −7.40098777849164017010301691281, −6.34230737393519977546136109524, −5.71457642807693882915289725747, −4.99685042203207454962302736572, −4.26621634984715969345986504863, −3.15465448478202026029024010476, −2.70999545264547105439037457450, −1.24926926559149769209483865152,
0.21474093403324632506536334603, 1.64448812096729895922813335251, 2.26871268935406146439087961326, 3.52150214121993301692533156308, 4.29939065640518268780094080336, 5.02208975585437185939165476127, 6.20584924047897577573336783450, 6.60657167291851586725206189153, 7.24566065186147106327968077487, 8.113009289605415314824151664137
