L(s) = 1 | + (0.292 + 1.70i)3-s − i·5-s − 3.41i·7-s + (−2.82 + i)9-s − 2.82·11-s + 2·13-s + (1.70 − 0.292i)15-s − 7.65i·17-s + 2.82i·19-s + (5.82 − i)21-s − 7.41·23-s − 25-s + (−2.53 − 4.53i)27-s − 8i·29-s − 5.65i·31-s + ⋯ |
L(s) = 1 | + (0.169 + 0.985i)3-s − 0.447i·5-s − 1.29i·7-s + (−0.942 + 0.333i)9-s − 0.852·11-s + 0.554·13-s + (0.440 − 0.0756i)15-s − 1.85i·17-s + 0.648i·19-s + (1.27 − 0.218i)21-s − 1.54·23-s − 0.200·25-s + (−0.487 − 0.872i)27-s − 1.48i·29-s − 1.01i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.823912 - 0.694589i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.823912 - 0.694589i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.292 - 1.70i)T \) |
| 5 | \( 1 + iT \) |
good | 7 | \( 1 + 3.41iT - 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 7.65iT - 17T^{2} \) |
| 19 | \( 1 - 2.82iT - 19T^{2} \) |
| 23 | \( 1 + 7.41T + 23T^{2} \) |
| 29 | \( 1 + 8iT - 29T^{2} \) |
| 31 | \( 1 + 5.65iT - 31T^{2} \) |
| 37 | \( 1 - 0.343T + 37T^{2} \) |
| 41 | \( 1 + 2iT - 41T^{2} \) |
| 43 | \( 1 - 7.89iT - 43T^{2} \) |
| 47 | \( 1 - 6.24T + 47T^{2} \) |
| 53 | \( 1 + 3.65iT - 53T^{2} \) |
| 59 | \( 1 - 1.65T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 + 1.07iT - 67T^{2} \) |
| 71 | \( 1 + 1.17T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + 4.48iT - 79T^{2} \) |
| 83 | \( 1 - 5.07T + 83T^{2} \) |
| 89 | \( 1 - 7.31iT - 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
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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
−9.869113963538283544560788599757, −9.263786899534983085947992525334, −7.968442973873206179526497107237, −7.72685636742223541199245844245, −6.26216519437737281059893779057, −5.30510685353685002044739947340, −4.38368658183573462338288319609, −3.74522112318939901511423993579, −2.45284117061914577248896877768, −0.47026160017617460907234407445,
1.73612211265992309580400064701, 2.60689882874429319990856351568, 3.66182829044067056772071800503, 5.33476764644705104083583278791, 6.00001014784392624777439472446, 6.73000673676486891701030068145, 7.82394939994925704118089280682, 8.477999427876743863663141121432, 9.060061852513993294726934255994, 10.41545090147013540728386669210