L(s) = 1 | + (0.0749 + 5.19i)3-s + (5.37 − 5.37i)5-s + 14.8·7-s + (−26.9 + 0.779i)9-s + (−30.0 − 30.0i)11-s + (−61.5 + 61.5i)13-s + (28.3 + 27.5i)15-s + 48.8i·17-s + (−7.45 − 7.45i)19-s + (1.11 + 77.1i)21-s + 43.0i·23-s + 67.1i·25-s + (−6.07 − 140. i)27-s + (32.9 + 32.9i)29-s + 173. i·31-s + ⋯ |
L(s) = 1 | + (0.0144 + 0.999i)3-s + (0.480 − 0.480i)5-s + 0.802·7-s + (−0.999 + 0.0288i)9-s + (−0.823 − 0.823i)11-s + (−1.31 + 1.31i)13-s + (0.487 + 0.473i)15-s + 0.696i·17-s + (−0.0900 − 0.0900i)19-s + (0.0115 + 0.802i)21-s + 0.390i·23-s + 0.537i·25-s + (−0.0432 − 0.999i)27-s + (0.211 + 0.211i)29-s + 1.00i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.035573437\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.035573437\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.0749 - 5.19i)T \) |
good | 5 | \( 1 + (-5.37 + 5.37i)T - 125iT^{2} \) |
| 7 | \( 1 - 14.8T + 343T^{2} \) |
| 11 | \( 1 + (30.0 + 30.0i)T + 1.33e3iT^{2} \) |
| 13 | \( 1 + (61.5 - 61.5i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 - 48.8iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (7.45 + 7.45i)T + 6.85e3iT^{2} \) |
| 23 | \( 1 - 43.0iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-32.9 - 32.9i)T + 2.43e4iT^{2} \) |
| 31 | \( 1 - 173. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-177. - 177. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + 454.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (239. - 239. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 - 30.4T + 1.03e5T^{2} \) |
| 53 | \( 1 + (235. - 235. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (260. + 260. i)T + 2.05e5iT^{2} \) |
| 61 | \( 1 + (-388. + 388. i)T - 2.26e5iT^{2} \) |
| 67 | \( 1 + (334. + 334. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 + 522. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 689. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 692. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-677. + 677. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 - 261.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 641.T + 9.12e5T^{2} \) |
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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
−11.24472030239862043779308568894, −10.34256565910285513332660691048, −9.515115802964424403325066434121, −8.697099239021215249640284870641, −7.85987213043529846231179974286, −6.37719147159962310342305283877, −5.09817857223411102858672941557, −4.75128505596515950948047720950, −3.23793526973716845022498484684, −1.79346621024801259932743639234,
0.32207974360259424830272712137, 2.08111400293661153022277993288, 2.77854012364626499286025248706, 4.84386603808809992004794127938, 5.64347225774835630774863353240, 6.90880962498991114600680948411, 7.64854255575825313804420925583, 8.316652060161773793561724621350, 9.790935127848789410010201339353, 10.45730326501878284030413480701