L(s) = 1 | + (−1.40 − 0.171i)2-s + 3-s + (1.94 + 0.481i)4-s − 0.963i·5-s + (−1.40 − 0.171i)6-s + (−2.64 − 1.00i)8-s + 9-s + (−0.165 + 1.35i)10-s + 5.48i·11-s + (1.94 + 0.481i)12-s + 3.75i·13-s − 0.963i·15-s + (3.53 + 1.87i)16-s − 0.686i·17-s + (−1.40 − 0.171i)18-s + 4.88·19-s + ⋯ |
L(s) = 1 | + (−0.992 − 0.121i)2-s + 0.577·3-s + (0.970 + 0.240i)4-s − 0.430i·5-s + (−0.573 − 0.0700i)6-s + (−0.934 − 0.356i)8-s + 0.333·9-s + (−0.0523 + 0.427i)10-s + 1.65i·11-s + (0.560 + 0.139i)12-s + 1.04i·13-s − 0.248i·15-s + (0.883 + 0.467i)16-s − 0.166i·17-s + (−0.330 − 0.0404i)18-s + 1.12·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.13782 + 0.272665i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13782 + 0.272665i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.40 + 0.171i)T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 0.963iT - 5T^{2} \) |
| 11 | \( 1 - 5.48iT - 11T^{2} \) |
| 13 | \( 1 - 3.75iT - 13T^{2} \) |
| 17 | \( 1 + 0.686iT - 17T^{2} \) |
| 19 | \( 1 - 4.88T + 19T^{2} \) |
| 23 | \( 1 - 1.24iT - 23T^{2} \) |
| 29 | \( 1 + 2.48T + 29T^{2} \) |
| 31 | \( 1 - 4.82T + 31T^{2} \) |
| 37 | \( 1 + 2.73T + 37T^{2} \) |
| 41 | \( 1 + 9.42iT - 41T^{2} \) |
| 43 | \( 1 - 5.97iT - 43T^{2} \) |
| 47 | \( 1 - 3.61T + 47T^{2} \) |
| 53 | \( 1 + 4.09T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 + 10.4iT - 61T^{2} \) |
| 67 | \( 1 - 9.43iT - 67T^{2} \) |
| 71 | \( 1 - 10.1iT - 71T^{2} \) |
| 73 | \( 1 + 6.66iT - 73T^{2} \) |
| 79 | \( 1 + 1.41iT - 79T^{2} \) |
| 83 | \( 1 - 0.543T + 83T^{2} \) |
| 89 | \( 1 + 0.554iT - 89T^{2} \) |
| 97 | \( 1 - 10.8iT - 97T^{2} \) |
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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
−10.47509071669950531862445650615, −9.575920017911069407517132220476, −9.240079386768438403700079383652, −8.224516549763564848611928743861, −7.29459418441137993104152327899, −6.75675769060298425379762003309, −5.16580309750718699638249464647, −3.96858590985942636688055683474, −2.53335899746608145265995416723, −1.45527940591372810576326316697,
0.953702447414658018111453212768, 2.75096565894104182827875493047, 3.40030963283269115989687941772, 5.38143639480652192837944215847, 6.28746476179350772537779380817, 7.30697496163011732771012659885, 8.173343855923604916632624491194, 8.709961405462273815248260459380, 9.721007480135429067734607025295, 10.52725941699851393985361801178