| L(s) = 1 | + (1.41 − 1.73i)5-s − 2.44·7-s + 1.41i·11-s − 2.44·13-s − 3.46·17-s + 6·19-s + 6.92i·23-s + (−0.999 − 4.89i)25-s − 5.65·29-s − 2i·31-s + (−3.46 + 4.24i)35-s − 7.34·37-s − 4.24i·41-s + 4.89i·43-s − 3.46i·47-s + ⋯ |
| L(s) = 1 | + (0.632 − 0.774i)5-s − 0.925·7-s + 0.426i·11-s − 0.679·13-s − 0.840·17-s + 1.37·19-s + 1.44i·23-s + (−0.199 − 0.979i)25-s − 1.05·29-s − 0.359i·31-s + (−0.585 + 0.717i)35-s − 1.20·37-s − 0.662i·41-s + 0.747i·43-s − 0.505i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4318530116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4318530116\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.41 + 1.73i)T \) |
| good | 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 6.92iT - 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 + 7.34T + 37T^{2} \) |
| 41 | \( 1 + 4.24iT - 41T^{2} \) |
| 43 | \( 1 - 4.89iT - 43T^{2} \) |
| 47 | \( 1 + 3.46iT - 47T^{2} \) |
| 53 | \( 1 + 6.92iT - 53T^{2} \) |
| 59 | \( 1 - 1.41iT - 59T^{2} \) |
| 61 | \( 1 - 12iT - 61T^{2} \) |
| 67 | \( 1 - 14.6iT - 67T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 - 9.79iT - 73T^{2} \) |
| 79 | \( 1 - 2iT - 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 4.24iT - 89T^{2} \) |
| 97 | \( 1 + 4.89iT - 97T^{2} \) |
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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
−9.201028674042013104539539641490, −8.451574040619934859724432314473, −7.32406384584604010542540810283, −6.95707268697914497651916910814, −5.73826278229352478940394256265, −5.39948692449473751146483524226, −4.38829546476684810241818369165, −3.44724091023850918827552373022, −2.40668547404895229682789301321, −1.37962018562395288118061553738,
0.12977991383724321686711533907, 1.83985426438235469516644842435, 2.88116437707920679438528231577, 3.39691999398368832798804146311, 4.63980756307150513347269347283, 5.55183867995277676508363770137, 6.30859718692458813845281736721, 6.89817246175767912643235720996, 7.54938011411892900667436721941, 8.630931229511793315636686270793
