L(s) = 1 | + (−2.07 + 0.839i)5-s + 4.93i·7-s − 2.41·11-s − 2.90i·13-s + 6.86i·17-s + 4.17·19-s − 3.35i·23-s + (3.58 − 3.48i)25-s − 5.19·29-s − 6.17·31-s + (−4.14 − 10.2i)35-s + 7.84i·37-s − 5.87·41-s − 4.93i·43-s − 11.9i·47-s + ⋯ |
L(s) = 1 | + (−0.926 + 0.375i)5-s + 1.86i·7-s − 0.727·11-s − 0.806i·13-s + 1.66i·17-s + 0.958·19-s − 0.700i·23-s + (0.717 − 0.696i)25-s − 0.964·29-s − 1.10·31-s + (−0.700 − 1.72i)35-s + 1.28i·37-s − 0.917·41-s − 0.752i·43-s − 1.73i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3546045881\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3546045881\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.07 - 0.839i)T \) |
good | 7 | \( 1 - 4.93iT - 7T^{2} \) |
| 11 | \( 1 + 2.41T + 11T^{2} \) |
| 13 | \( 1 + 2.90iT - 13T^{2} \) |
| 17 | \( 1 - 6.86iT - 17T^{2} \) |
| 19 | \( 1 - 4.17T + 19T^{2} \) |
| 23 | \( 1 + 3.35iT - 23T^{2} \) |
| 29 | \( 1 + 5.19T + 29T^{2} \) |
| 31 | \( 1 + 6.17T + 31T^{2} \) |
| 37 | \( 1 - 7.84iT - 37T^{2} \) |
| 41 | \( 1 + 5.87T + 41T^{2} \) |
| 43 | \( 1 + 4.93iT - 43T^{2} \) |
| 47 | \( 1 + 11.9iT - 47T^{2} \) |
| 53 | \( 1 + 8.54iT - 53T^{2} \) |
| 59 | \( 1 - 1.05T + 59T^{2} \) |
| 61 | \( 1 - 9.17T + 61T^{2} \) |
| 67 | \( 1 - 4.05iT - 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 2.02iT - 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - 5.18iT - 83T^{2} \) |
| 89 | \( 1 + 3.09T + 89T^{2} \) |
| 97 | \( 1 - 0.882iT - 97T^{2} \) |
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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
−9.939391991144492713393993094802, −8.655781595214613705793037668694, −8.449191376182983529901601915458, −7.60864306265519890222196946067, −6.59976881265957756345191929692, −5.60959662488680100772274451667, −5.16762367874670730463316849582, −3.74373155189888670241321788738, −2.98677255989246034600843760227, −1.97695313580484505490615316462,
0.14230163989165245450413356432, 1.31281490460219658780685591310, 3.07007998573965699000917187943, 3.92270048991999864574954141683, 4.63115388943308374742408299521, 5.46189977023397955555606576692, 6.95320023034237060159973963686, 7.47551538443078409144176124243, 7.73846503294124941876002776535, 9.108835724102660546261536714178