L(s) = 1 | + (−2.72 − 0.773i)2-s + (6.80 + 4.21i)4-s + 20.8i·5-s − 13.9i·7-s + (−15.2 − 16.7i)8-s + (16.1 − 56.7i)10-s − 34.5·11-s − 31.3·13-s + (−10.7 + 37.8i)14-s + (28.5 + 57.2i)16-s − 34.4i·17-s + 120. i·19-s + (−87.7 + 141. i)20-s + (93.8 + 26.7i)22-s − 137.·23-s + ⋯ |
L(s) = 1 | + (−0.961 − 0.273i)2-s + (0.850 + 0.526i)4-s + 1.86i·5-s − 0.750i·7-s + (−0.673 − 0.738i)8-s + (0.510 − 1.79i)10-s − 0.945·11-s − 0.668·13-s + (−0.205 + 0.722i)14-s + (0.445 + 0.895i)16-s − 0.491i·17-s + 1.45i·19-s + (−0.981 + 1.58i)20-s + (0.909 + 0.258i)22-s − 1.24·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.120282 + 0.422795i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.120282 + 0.422795i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.72 + 0.773i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 20.8iT - 125T^{2} \) |
| 7 | \( 1 + 13.9iT - 343T^{2} \) |
| 11 | \( 1 + 34.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 31.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 34.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 120. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 137.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 93.1iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 111. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 146.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 8.44iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 427. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 318.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 291. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 364.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 289.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 305. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 102.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 442.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 245. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 478.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.41e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
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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
−13.81695595146035647867261865578, −12.27917939197709249807105203494, −11.16337224487141225208746933801, −10.34550443990491264556600663211, −9.862087619822094309370609722745, −7.87752696595028383643677735288, −7.32405486742889931902126593435, −6.15473072863399169093223001726, −3.57949107565351646971696693997, −2.33707485222832699916042660376,
0.30432762571761501793149754270, 2.12991481930781626514433842052, 4.87981700063386842384458082735, 5.80694550585401314966782390789, 7.60522181874710645815746440140, 8.606778924466525929943087186890, 9.217803647424019148046820922053, 10.38131376855891137862894583246, 11.89255195765867715148052979368, 12.51273512383937757227258966451