L(s) = 1 | + (−3.62 + 3.72i)3-s + (0.263 + 0.456i)5-s + (16.6 + 8.05i)7-s + (−0.685 − 26.9i)9-s + (9.84 + 5.68i)11-s − 65.8i·13-s + (−2.65 − 0.674i)15-s + (0.339 − 0.587i)17-s + (19.0 − 10.9i)19-s + (−90.4 + 32.8i)21-s + (6.60 − 3.81i)23-s + (62.3 − 108. i)25-s + (102. + 95.3i)27-s + 165. i·29-s + (110. + 63.6i)31-s + ⋯ |
L(s) = 1 | + (−0.698 + 0.716i)3-s + (0.0235 + 0.0407i)5-s + (0.900 + 0.435i)7-s + (−0.0253 − 0.999i)9-s + (0.269 + 0.155i)11-s − 1.40i·13-s + (−0.0456 − 0.0116i)15-s + (0.00483 − 0.00837i)17-s + (0.229 − 0.132i)19-s + (−0.940 + 0.340i)21-s + (0.0598 − 0.0345i)23-s + (0.498 − 0.864i)25-s + (0.733 + 0.679i)27-s + 1.06i·29-s + (0.639 + 0.369i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 - 0.456i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.889 - 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.642159001\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.642159001\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (3.62 - 3.72i)T \) |
| 7 | \( 1 + (-16.6 - 8.05i)T \) |
good | 5 | \( 1 + (-0.263 - 0.456i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-9.84 - 5.68i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 65.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-0.339 + 0.587i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-19.0 + 10.9i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-6.60 + 3.81i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 165. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-110. - 63.6i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (95.9 + 166. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 508.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 25.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-167. - 289. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-376. - 217. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (208. - 361. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (288. - 166. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-68.6 + 118. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 718. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-1.00e3 - 582. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (189. + 327. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 539.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-457. - 792. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 29.2iT - 9.12e5T^{2} \) |
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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
−10.93062340549559556074318246200, −10.57728470070032019974740352885, −9.388058949862083049438184851043, −8.505159095006370457817369794528, −7.38061506764795848458285992353, −6.03903270897564138116624941930, −5.24242619317812302102158473472, −4.30174281474544583651894906239, −2.85231436302586988029271690924, −0.921886128898117008419042879795,
0.969550684893816956663191660216, 2.11831327944877487892425206733, 4.12539194086943405028136577118, 5.11627596010266784780715784645, 6.28360609609356344925770082210, 7.17605986474935913633904160954, 8.023809188329207019695701222025, 9.126713265804667393752274107747, 10.34239407844741535887453975061, 11.43110922682190536512942146400