| L(s) = 1 | + 2.16i·3-s + 8.18·7-s + 4.29·9-s + 3.30·11-s − 8.86i·13-s + 10.2·17-s + (18.5 − 4.03i)19-s + 17.7i·21-s + 0.896·23-s + 28.8i·27-s + 17.9i·29-s − 40.8i·31-s + 7.16i·33-s − 7.68i·37-s + 19.2·39-s + ⋯ |
| L(s) = 1 | + 0.723i·3-s + 1.16·7-s + 0.477·9-s + 0.300·11-s − 0.681i·13-s + 0.602·17-s + (0.977 − 0.212i)19-s + 0.845i·21-s + 0.0389·23-s + 1.06i·27-s + 0.617i·29-s − 1.31i·31-s + 0.217i·33-s − 0.207i·37-s + 0.492·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.212i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.809210004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.809210004\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-18.5 + 4.03i)T \) |
| good | 3 | \( 1 - 2.16iT - 9T^{2} \) |
| 7 | \( 1 - 8.18T + 49T^{2} \) |
| 11 | \( 1 - 3.30T + 121T^{2} \) |
| 13 | \( 1 + 8.86iT - 169T^{2} \) |
| 17 | \( 1 - 10.2T + 289T^{2} \) |
| 23 | \( 1 - 0.896T + 529T^{2} \) |
| 29 | \( 1 - 17.9iT - 841T^{2} \) |
| 31 | \( 1 + 40.8iT - 961T^{2} \) |
| 37 | \( 1 + 7.68iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 69.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 25.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 12.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 55.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 105. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 48.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 14.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 2.89iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 75.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 86.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 52.0T + 6.88e3T^{2} \) |
| 89 | \( 1 - 39.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 172. iT - 9.40e3T^{2} \) |
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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
−9.264260154101417271199790941824, −8.171804693274567271318487419404, −7.66385357565233382952950459128, −6.78309269848037650249556600152, −5.48203448321915098624626756312, −5.08877359539709281613743790189, −4.11368119816744380875389610745, −3.35584019941126587672616152095, −1.99946852359367986671862442715, −0.880301223724119926935084084635,
1.14853996095099549278966117744, 1.65159220090149149076340309329, 2.93012428958855054621155837408, 4.19409075085847301298815289237, 4.87809342607020692805510382640, 5.88663400416101260790595196967, 6.75705168934825532811691240526, 7.54647856679899693577636666050, 7.988694167715708632232919823199, 8.923427621675197530640019491373
