L(s) = 1 | − i·3-s + (−2.20 − 0.353i)5-s − 1.65i·7-s − 9-s − 2.94·11-s + i·13-s + (−0.353 + 2.20i)15-s + 1.46i·17-s − 1.65·21-s − 0.532i·23-s + (4.74 + 1.56i)25-s + i·27-s − 5.70·29-s + 2.94i·33-s + (−0.585 + 3.65i)35-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (−0.987 − 0.158i)5-s − 0.625i·7-s − 0.333·9-s − 0.888·11-s + 0.277i·13-s + (−0.0913 + 0.570i)15-s + 0.355i·17-s − 0.361·21-s − 0.111i·23-s + (0.949 + 0.312i)25-s + 0.192i·27-s − 1.06·29-s + 0.513i·33-s + (−0.0989 + 0.617i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5091734911\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5091734911\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (2.20 + 0.353i)T \) |
| 13 | \( 1 - iT \) |
good | 7 | \( 1 + 1.65iT - 7T^{2} \) |
| 11 | \( 1 + 2.94T + 11T^{2} \) |
| 17 | \( 1 - 1.46iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 0.532iT - 23T^{2} \) |
| 29 | \( 1 + 5.70T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 8.77iT - 37T^{2} \) |
| 41 | \( 1 - 1.23T + 41T^{2} \) |
| 43 | \( 1 - 1.70iT - 43T^{2} \) |
| 47 | \( 1 - 2.70iT - 47T^{2} \) |
| 53 | \( 1 - 8.77iT - 53T^{2} \) |
| 59 | \( 1 - 3.83T + 59T^{2} \) |
| 61 | \( 1 + 0.241T + 61T^{2} \) |
| 67 | \( 1 - 2.58iT - 67T^{2} \) |
| 71 | \( 1 + 2.55T + 71T^{2} \) |
| 73 | \( 1 + 0.188iT - 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 - 7.91iT - 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 - 16.8iT - 97T^{2} \) |
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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
−9.570464018123416972949316395033, −8.578530172099644741760716897644, −7.911769787457811455367846601904, −7.35893864226951915312736743114, −6.58110484223890047343384518172, −5.50052427995353639584291080788, −4.53428465509764037398173187037, −3.68103080528412529963105124144, −2.61306449419521224583889078974, −1.16960671105137827370954718128,
0.21858177390418728187093117875, 2.31420867604760437730410646515, 3.27967583110878382253294619736, 4.12464807928642992270040862500, 5.14494671945012328141402063142, 5.74555070725823687404137994974, 7.01426009182304390297347416298, 7.73148144955919080357355233839, 8.479101107061021221173788106718, 9.199128288021881850708734928207