L(s) = 1 | + (−1.69 − 0.348i)3-s − 5-s + (2.63 − 0.272i)7-s + (2.75 + 1.18i)9-s − 3.13i·11-s + 4.34i·13-s + (1.69 + 0.348i)15-s + 0.791·17-s + 0.768i·19-s + (−4.56 − 0.453i)21-s − 6.32i·23-s + 25-s + (−4.26 − 2.96i)27-s + 5.96i·29-s + 5.49i·31-s + ⋯ |
L(s) = 1 | + (−0.979 − 0.201i)3-s − 0.447·5-s + (0.994 − 0.103i)7-s + (0.919 + 0.393i)9-s − 0.944i·11-s + 1.20i·13-s + (0.438 + 0.0899i)15-s + 0.191·17-s + 0.176i·19-s + (−0.995 − 0.0990i)21-s − 1.31i·23-s + 0.200·25-s + (−0.821 − 0.570i)27-s + 1.10i·29-s + 0.986i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.239986319\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.239986319\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.69 + 0.348i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-2.63 + 0.272i)T \) |
good | 11 | \( 1 + 3.13iT - 11T^{2} \) |
| 13 | \( 1 - 4.34iT - 13T^{2} \) |
| 17 | \( 1 - 0.791T + 17T^{2} \) |
| 19 | \( 1 - 0.768iT - 19T^{2} \) |
| 23 | \( 1 + 6.32iT - 23T^{2} \) |
| 29 | \( 1 - 5.96iT - 29T^{2} \) |
| 31 | \( 1 - 5.49iT - 31T^{2} \) |
| 37 | \( 1 - 1.83T + 37T^{2} \) |
| 41 | \( 1 - 1.60T + 41T^{2} \) |
| 43 | \( 1 + 6.18T + 43T^{2} \) |
| 47 | \( 1 - 3.24T + 47T^{2} \) |
| 53 | \( 1 - 10.0iT - 53T^{2} \) |
| 59 | \( 1 - 9.30T + 59T^{2} \) |
| 61 | \( 1 + 13.1iT - 61T^{2} \) |
| 67 | \( 1 - 2.64T + 67T^{2} \) |
| 71 | \( 1 - 0.227iT - 71T^{2} \) |
| 73 | \( 1 + 15.7iT - 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 2.74T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 - 6.11iT - 97T^{2} \) |
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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
−9.228362996682584832903996525096, −8.478996574981421450164167935099, −7.71239486612549479486106650139, −6.83950623340773851732037563039, −6.20511650343566924229022272432, −5.10791612582172434768967690324, −4.58585322717907997049738371414, −3.56203555908829874574377220860, −1.99360250705498550778767276095, −0.857784875094917908575325642508,
0.812967839232920386039937587302, 2.13798353847997291396752862063, 3.66724379407905797008159065201, 4.51775776599421287297757476411, 5.28141270841859792580860565193, 5.90436153273061435295438966074, 7.12308119968285738059474599853, 7.65942084485594622828952235758, 8.413195240951946210098121571926, 9.665274320095500440956911089657