| L(s) = 1 | + (0.167 + 0.818i)2-s + (−0.987 − 0.160i)3-s + (1.19 − 0.510i)4-s + (−2.79 + 0.688i)5-s + (−0.0336 − 0.834i)6-s + (0.263 − 0.555i)7-s + (1.56 + 2.26i)8-s + (0.948 + 0.316i)9-s + (−1.02 − 2.17i)10-s + (0.959 − 0.320i)11-s + (−1.26 + 0.311i)12-s + (2.34 + 2.73i)13-s + (0.498 + 0.122i)14-s + (2.86 − 0.231i)15-s + (0.209 − 0.217i)16-s + (−0.927 + 1.95i)17-s + ⋯ |
| L(s) = 1 | + (0.118 + 0.578i)2-s + (−0.569 − 0.0926i)3-s + (0.599 − 0.255i)4-s + (−1.24 + 0.307i)5-s + (−0.0137 − 0.340i)6-s + (0.0995 − 0.209i)7-s + (0.553 + 0.802i)8-s + (0.316 + 0.105i)9-s + (−0.325 − 0.686i)10-s + (0.289 − 0.0965i)11-s + (−0.365 + 0.0899i)12-s + (0.651 + 0.758i)13-s + (0.133 + 0.0328i)14-s + (0.740 − 0.0597i)15-s + (0.0523 − 0.0544i)16-s + (−0.225 + 0.474i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0283 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0283 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.870442 + 0.846078i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.870442 + 0.846078i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.987 + 0.160i)T \) |
| 13 | \( 1 + (-2.34 - 2.73i)T \) |
| good | 2 | \( 1 + (-0.167 - 0.818i)T + (-1.83 + 0.783i)T^{2} \) |
| 5 | \( 1 + (2.79 - 0.688i)T + (4.42 - 2.32i)T^{2} \) |
| 7 | \( 1 + (-0.263 + 0.555i)T + (-4.42 - 5.42i)T^{2} \) |
| 11 | \( 1 + (-0.959 + 0.320i)T + (8.79 - 6.60i)T^{2} \) |
| 17 | \( 1 + (0.927 - 1.95i)T + (-10.7 - 13.1i)T^{2} \) |
| 19 | \( 1 + (-2.94 - 5.09i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.03 - 5.25i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.181 - 0.890i)T + (-26.6 + 11.3i)T^{2} \) |
| 31 | \( 1 + (0.447 + 0.235i)T + (17.6 + 25.5i)T^{2} \) |
| 37 | \( 1 + (-4.65 + 2.94i)T + (15.8 - 33.4i)T^{2} \) |
| 41 | \( 1 + (-7.92 - 1.28i)T + (38.8 + 12.9i)T^{2} \) |
| 43 | \( 1 + (1.11 + 0.704i)T + (18.4 + 38.8i)T^{2} \) |
| 47 | \( 1 + (-1.28 - 10.6i)T + (-45.6 + 11.2i)T^{2} \) |
| 53 | \( 1 + (7.36 + 10.6i)T + (-18.7 + 49.5i)T^{2} \) |
| 59 | \( 1 + (4.52 + 4.71i)T + (-2.37 + 58.9i)T^{2} \) |
| 61 | \( 1 + (9.59 + 0.774i)T + (60.2 + 9.78i)T^{2} \) |
| 67 | \( 1 + (2.18 + 0.932i)T + (46.4 + 48.3i)T^{2} \) |
| 71 | \( 1 + (-7.39 + 9.06i)T + (-14.2 - 69.5i)T^{2} \) |
| 73 | \( 1 + (-0.322 - 0.285i)T + (8.79 + 72.4i)T^{2} \) |
| 79 | \( 1 + (1.60 + 13.2i)T + (-76.7 + 18.9i)T^{2} \) |
| 83 | \( 1 + (2.69 - 7.11i)T + (-62.1 - 55.0i)T^{2} \) |
| 89 | \( 1 + (-5.56 + 9.63i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.94 - 6.71i)T + (-81.9 + 51.8i)T^{2} \) |
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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
−11.27590719104094552641924487382, −10.58172029861846664552590203245, −9.341731808737982120081504544802, −7.80702878338502796456060013086, −7.66900184518041631602674616444, −6.45429009629615203565152863351, −5.85364752388849460307554115196, −4.48266654284089191005420932449, −3.52178769723722769885498726534, −1.56753258632673162991884731728,
0.810369552681124508921878929138, 2.69905238246722724249855124505, 3.86170525900845245438713476856, 4.70618813897292865058496378008, 6.08283411052950097619475048604, 7.13350778126175629893284971482, 7.88123749632698833884598239312, 8.893596636469926235746977254516, 10.13835268249345454556382080614, 11.04904161940328382496024017001
