| L(s) = 1 | − i·2-s + 4-s + (−1 + 2i)5-s − 2i·7-s − 3i·8-s + (2 + i)10-s − 4i·13-s − 2·14-s − 16-s − 2i·17-s + (−1 + 2i)20-s + 2i·23-s + (−3 − 4i)25-s − 4·26-s − 2i·28-s + 29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.5·4-s + (−0.447 + 0.894i)5-s − 0.755i·7-s − 1.06i·8-s + (0.632 + 0.316i)10-s − 1.10i·13-s − 0.534·14-s − 0.250·16-s − 0.485i·17-s + (−0.223 + 0.447i)20-s + 0.417i·23-s + (−0.600 − 0.800i)25-s − 0.784·26-s − 0.377i·28-s + 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.592519615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.592519615\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (1 - 2i)T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + iT - 2T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 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
−10.00505846949518651350295593141, −8.485499092341623796456950824390, −7.60242132995005891459753887582, −7.00451176912474970652342581349, −6.28641752581960476903018400151, −5.03942382321404709281914026419, −3.71728796090219176312261600033, −3.23711203771078452186518527739, −2.17323644387502277931074269001, −0.65504233145492505097997711043,
1.55265502428620737703863827011, 2.67440266786593693114220106013, 4.10809068241645342297028270241, 4.98732694313822228938769051418, 5.84487695249309432141090844679, 6.59130344346890757875140372092, 7.46510970093759013940925842897, 8.358543160738460753766710413005, 8.773355966290434546116607870882, 9.673208401878116675694158808760
