| L(s) = 1 | + 0.517·3-s + (−3.34 + 3.34i)7-s − 2.73·9-s + (1.09 + 1.09i)11-s − 4.89i·13-s + (0.707 − 0.707i)17-s + (2.09 + 2.09i)19-s + (−1.73 + 1.73i)21-s + (−4.38 − 4.38i)23-s − 2.96·27-s + (4.73 − 4.73i)29-s − 6.19i·31-s + (0.568 + 0.568i)33-s − 6.03i·37-s − 2.53i·39-s + ⋯ |
| L(s) = 1 | + 0.298·3-s + (−1.26 + 1.26i)7-s − 0.910·9-s + (0.331 + 0.331i)11-s − 1.35i·13-s + (0.171 − 0.171i)17-s + (0.481 + 0.481i)19-s + (−0.377 + 0.377i)21-s + (−0.913 − 0.913i)23-s − 0.571·27-s + (0.878 − 0.878i)29-s − 1.11i·31-s + (0.0989 + 0.0989i)33-s − 0.992i·37-s − 0.406i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7124681855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7124681855\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 0.517T + 3T^{2} \) |
| 7 | \( 1 + (3.34 - 3.34i)T - 7iT^{2} \) |
| 11 | \( 1 + (-1.09 - 1.09i)T + 11iT^{2} \) |
| 13 | \( 1 + 4.89iT - 13T^{2} \) |
| 17 | \( 1 + (-0.707 + 0.707i)T - 17iT^{2} \) |
| 19 | \( 1 + (-2.09 - 2.09i)T + 19iT^{2} \) |
| 23 | \( 1 + (4.38 + 4.38i)T + 23iT^{2} \) |
| 29 | \( 1 + (-4.73 + 4.73i)T - 29iT^{2} \) |
| 31 | \( 1 + 6.19iT - 31T^{2} \) |
| 37 | \( 1 + 6.03iT - 37T^{2} \) |
| 41 | \( 1 - 0.464iT - 41T^{2} \) |
| 43 | \( 1 + 0.656iT - 43T^{2} \) |
| 47 | \( 1 + (-1.41 - 1.41i)T + 47iT^{2} \) |
| 53 | \( 1 + 9.89T + 53T^{2} \) |
| 59 | \( 1 + (-7.73 + 7.73i)T - 59iT^{2} \) |
| 61 | \( 1 + (3.19 + 3.19i)T + 61iT^{2} \) |
| 67 | \( 1 - 5.79iT - 67T^{2} \) |
| 71 | \( 1 + 0.928T + 71T^{2} \) |
| 73 | \( 1 + (-8.81 + 8.81i)T - 73iT^{2} \) |
| 79 | \( 1 + 2.19T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 + (11.5 - 11.5i)T - 97iT^{2} \) |
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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
−9.315399297156348153443242608097, −8.315602886437310353850521582585, −7.87457936146157150331420541170, −6.51674902516764965490310091478, −5.93149757331048099761749500121, −5.31592428083177540033449470080, −3.87096417715926125811682693501, −2.93564354201617942627332695495, −2.35658496266480225557924890000, −0.26762437755117580598671186769,
1.32629707964591237332481857592, 2.92920417682380723636675322398, 3.57544106681101649733249293270, 4.45063394965528265472448403065, 5.68498998042055308981158946318, 6.64376674418476194361075419576, 7.02136160262405885559719073655, 8.104112631130487397293933791277, 8.956303923097829700680032070379, 9.590247782391737737203579417042
