L(s) = 1 | + (1.70 + 0.323i)3-s − 1.55i·5-s + (2.79 + 1.09i)9-s − 2.53·13-s + (0.502 − 2.64i)15-s − 4.33i·17-s − 6.83i·19-s − 3.80·23-s + 2.58·25-s + (4.39 + 2.77i)27-s − 8.33i·29-s + 4.89i·31-s + 1.58·37-s + (−4.31 − 0.818i)39-s − 7.44i·41-s + ⋯ |
L(s) = 1 | + (0.982 + 0.186i)3-s − 0.695i·5-s + (0.930 + 0.366i)9-s − 0.702·13-s + (0.129 − 0.683i)15-s − 1.05i·17-s − 1.56i·19-s − 0.794·23-s + 0.516·25-s + (0.845 + 0.533i)27-s − 1.54i·29-s + 0.879i·31-s + 0.260·37-s + (−0.690 − 0.131i)39-s − 1.16i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.326964801\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.326964801\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.70 - 0.323i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 1.55iT - 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2.53T + 13T^{2} \) |
| 17 | \( 1 + 4.33iT - 17T^{2} \) |
| 19 | \( 1 + 6.83iT - 19T^{2} \) |
| 23 | \( 1 + 3.80T + 23T^{2} \) |
| 29 | \( 1 + 8.33iT - 29T^{2} \) |
| 31 | \( 1 - 4.89iT - 31T^{2} \) |
| 37 | \( 1 - 1.58T + 37T^{2} \) |
| 41 | \( 1 + 7.44iT - 41T^{2} \) |
| 43 | \( 1 + 6.20iT - 43T^{2} \) |
| 47 | \( 1 + 5.38T + 47T^{2} \) |
| 53 | \( 1 - 10.5iT - 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 8.19T + 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 - 6.92iT - 79T^{2} \) |
| 83 | \( 1 - 4.82T + 83T^{2} \) |
| 89 | \( 1 + 4.33iT - 89T^{2} \) |
| 97 | \( 1 - 9.89T + 97T^{2} \) |
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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
−8.834740752828440960292984838401, −8.242761540445103312941461415785, −7.31887462199056344876980401136, −6.82631192015691729782654896872, −5.42966447494770659242717338679, −4.74621372862435785130181442555, −4.04459518865902132337967175876, −2.85502780801859980337389406750, −2.18163599066935038818789561074, −0.69173339477106989740752220423,
1.50507522720360483509853255266, 2.40742514637745053384540082463, 3.39019437622004944252738651733, 3.99952419429271343112706968292, 5.14600471242184446367329507139, 6.30125886935291410948641284755, 6.81386232618610164599568951206, 7.910338868549873558524737436221, 8.075776680033342679941905563429, 9.110308962676599777638448874554