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
−8.584789733169708100502998327582, −7.37058066298396852677211163843, −7.10561920981037500636512424788, −5.96501243686338160461523499620, −5.52287348461202459763915636707, −4.40197020553208986931036561129, −3.98375365161379524643313847903, −3.06226584785958620294373954251, −2.02222568598840384658072395382, −0.821697136972989003429041974865,
0.864893273934733304224739975593, 1.67354917069516370498979899422, 2.83347848572936620835717912989, 3.59871818034364159663470852085, 4.49928669266261078510024340149, 5.41497664961217517149867291293, 6.21237854204797796213611000442, 6.67566369797499408615472362577, 7.64713187145131429405954477914, 8.011278304171970635589209087344