| L(s) = 1 | − i·3-s − 9-s + 4·11-s − 2i·13-s + 2i·17-s + 4·19-s + i·27-s − 2·29-s − 4i·33-s + 10i·37-s − 2·39-s + 10·41-s + 4i·43-s + 8i·47-s + 7·49-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.333·9-s + 1.20·11-s − 0.554i·13-s + 0.485i·17-s + 0.917·19-s + 0.192i·27-s − 0.371·29-s − 0.696i·33-s + 1.64i·37-s − 0.320·39-s + 1.56·41-s + 0.609i·43-s + 1.16i·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.141583670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.141583670\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 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
−8.011278304171970635589209087344, −7.64713187145131429405954477914, −6.67566369797499408615472362577, −6.21237854204797796213611000442, −5.41497664961217517149867291293, −4.49928669266261078510024340149, −3.59871818034364159663470852085, −2.83347848572936620835717912989, −1.67354917069516370498979899422, −0.864893273934733304224739975593,
0.821697136972989003429041974865, 2.02222568598840384658072395382, 3.06226584785958620294373954251, 3.98375365161379524643313847903, 4.40197020553208986931036561129, 5.52287348461202459763915636707, 5.96501243686338160461523499620, 7.10561920981037500636512424788, 7.37058066298396852677211163843, 8.584789733169708100502998327582
