| L(s) = 1 | + 1.18i·2-s + (1.69 + 0.368i)3-s + 0.601·4-s + (−0.435 + 2.00i)6-s + 0.737·7-s + 3.07i·8-s + (2.72 + 1.24i)9-s − 3.06i·11-s + (1.01 + 0.221i)12-s + 0.872i·14-s − 2.43·16-s + (−1.47 + 3.22i)18-s − 2.36·19-s + (1.24 + 0.271i)21-s + 3.62·22-s + ⋯ |
| L(s) = 1 | + 0.836i·2-s + (0.977 + 0.212i)3-s + 0.300·4-s + (−0.177 + 0.817i)6-s + 0.278·7-s + 1.08i·8-s + (0.909 + 0.415i)9-s − 0.923i·11-s + (0.293 + 0.0639i)12-s + 0.233i·14-s − 0.609·16-s + (−0.347 + 0.760i)18-s − 0.543·19-s + (0.272 + 0.0593i)21-s + 0.772·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 501 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.212 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 501 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.212 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.78461 + 1.43796i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.78461 + 1.43796i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.69 - 0.368i)T \) |
| 167 | \( 1 - 12.9iT \) |
| good | 2 | \( 1 - 1.18iT - 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 0.737T + 7T^{2} \) |
| 11 | \( 1 + 3.06iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 2.36T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 3.12iT - 29T^{2} \) |
| 31 | \( 1 - 1.95T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 3.40iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 7.68T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 12.1iT - 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.04866954621158102714852957021, −10.14477237087609487765565097158, −9.027888397976937515359992223230, −8.235687290013756001016519714363, −7.69289718192450060543048025159, −6.62225083815674853794057337941, −5.68691535293988355281851013758, −4.46406118860174400084325698260, −3.17917760593913994421382836563, −1.97583335081593773170740865908,
1.56300288856658470828684855626, 2.46359617380663814998568503115, 3.60742330403517438764072303142, 4.60451613500789500615645451564, 6.33095256684705682881341439338, 7.24636645554559184974840564606, 8.028310720332244060642842995399, 9.178120672372703614203419546726, 9.900849461276931765592435515647, 10.65925391268766852862830671243
