L(s) = 1 | + (−0.146 − 0.989i)3-s + (−0.463 + 0.886i)5-s + (−0.929 − 0.368i)7-s + (−0.957 + 0.289i)9-s + (−0.913 − 0.406i)11-s + (−0.604 − 0.796i)13-s + (0.944 + 0.328i)15-s + (−0.895 − 0.444i)17-s + (0.0209 + 0.999i)19-s + (−0.228 + 0.973i)21-s + (−0.855 − 0.518i)23-s + (−0.570 − 0.821i)25-s + (0.425 + 0.904i)27-s + (−0.228 + 0.973i)29-s + (−0.957 + 0.289i)31-s + ⋯ |
L(s) = 1 | + (−0.146 − 0.989i)3-s + (−0.463 + 0.886i)5-s + (−0.929 − 0.368i)7-s + (−0.957 + 0.289i)9-s + (−0.913 − 0.406i)11-s + (−0.604 − 0.796i)13-s + (0.944 + 0.328i)15-s + (−0.895 − 0.444i)17-s + (0.0209 + 0.999i)19-s + (−0.228 + 0.973i)21-s + (−0.855 − 0.518i)23-s + (−0.570 − 0.821i)25-s + (0.425 + 0.904i)27-s + (−0.228 + 0.973i)29-s + (−0.957 + 0.289i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1606020146 - 0.08800499297i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1606020146 - 0.08800499297i\) |
\(L(1)\) |
\(\approx\) |
\(0.5094939238 - 0.1004961958i\) |
\(L(1)\) |
\(\approx\) |
\(0.5094939238 - 0.1004961958i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 \) |
good | 3 | \( 1 + (-0.146 - 0.989i)T \) |
| 5 | \( 1 + (-0.463 + 0.886i)T \) |
| 7 | \( 1 + (-0.929 - 0.368i)T \) |
| 11 | \( 1 + (-0.913 - 0.406i)T \) |
| 13 | \( 1 + (-0.604 - 0.796i)T \) |
| 17 | \( 1 + (-0.895 - 0.444i)T \) |
| 19 | \( 1 + (0.0209 + 0.999i)T \) |
| 23 | \( 1 + (-0.855 - 0.518i)T \) |
| 29 | \( 1 + (-0.228 + 0.973i)T \) |
| 31 | \( 1 + (-0.957 + 0.289i)T \) |
| 37 | \( 1 + (0.570 + 0.821i)T \) |
| 41 | \( 1 + (-0.809 + 0.587i)T \) |
| 43 | \( 1 + (-0.535 + 0.844i)T \) |
| 47 | \( 1 + (0.463 - 0.886i)T \) |
| 53 | \( 1 + (0.809 + 0.587i)T \) |
| 59 | \( 1 + (-0.985 - 0.166i)T \) |
| 61 | \( 1 + (-0.913 + 0.406i)T \) |
| 67 | \( 1 + (-0.228 + 0.973i)T \) |
| 71 | \( 1 + (-0.187 + 0.982i)T \) |
| 73 | \( 1 + (-0.5 - 0.866i)T \) |
| 79 | \( 1 + (-0.0209 + 0.999i)T \) |
| 83 | \( 1 + (-0.669 - 0.743i)T \) |
| 89 | \( 1 + (-0.999 + 0.0418i)T \) |
| 97 | \( 1 + (0.146 - 0.989i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.61165735567123072918929147920, −16.97932688599519688510769871851, −16.449513175105761871631157095215, −15.70907981636547141379935746924, −15.48630634572226531910657991042, −14.89033697306300428637015148142, −13.73094010283028947072395984083, −13.24862431337447363482200771393, −12.4385356347778287660643723501, −11.92857482499740943136961336163, −11.205334808835551216300778439952, −10.49182763302777522263346116548, −9.65173966687907112383777768060, −9.262988409574914659908692175100, −8.73446915125376886639434659471, −7.837420895706376776466973800235, −7.092055632082659677059131601545, −6.13239935344089564619190318988, −5.48506892132900346193246433532, −4.788384892242196992256002310326, −4.18602742634411405183306497308, −3.59351076367204726550346838781, −2.5585494881520143573925242737, −1.95723957654010215992927421863, −0.25019396064681777923996478326,
0.16500447369834685235186204384, 1.44056275961650716618232439047, 2.53904720435086838817703343218, 2.923965632870846942704594342277, 3.61328398633311431445949867628, 4.674254746945996177164691116527, 5.69441832614172005584502589587, 6.14961295591358553604764579632, 7.0153179569733946728368694144, 7.34438976591878112461228757211, 8.08568845640328166543905057448, 8.683509022098567855191553074261, 9.88387797593757804638548871687, 10.383247787846336415536576690775, 10.98041108558407362070518089624, 11.74613320086772812018783421803, 12.44016069060315462047400536517, 12.98606438347516979863132855796, 13.59344989220958963182484348380, 14.23675461633505914665810739658, 14.96229313019186895948507403395, 15.632410428711179255027788419754, 16.453180902056174051550517240525, 16.82249195158177560395350896661, 17.98613439569805853956531576059