| L(s) = 1 | + (−2.05 + 0.852i)2-s + (0.814 − 4.09i)3-s + (0.677 − 0.677i)4-s + (5.08 − 3.39i)5-s + (1.81 + 9.12i)6-s + (−2.63 − 1.76i)7-s + (2.59 − 6.25i)8-s + (−7.80 − 3.23i)9-s + (−7.56 + 11.3i)10-s + (20.7 − 4.12i)11-s + (−2.22 − 3.32i)12-s + (−0.501 − 0.501i)13-s + (6.92 + 1.37i)14-s + (−9.77 − 23.6i)15-s + 18.9i·16-s + ⋯ |
| L(s) = 1 | + (−1.02 + 0.426i)2-s + (0.271 − 1.36i)3-s + (0.169 − 0.169i)4-s + (1.01 − 0.679i)5-s + (0.302 + 1.52i)6-s + (−0.376 − 0.251i)7-s + (0.324 − 0.782i)8-s + (−0.866 − 0.359i)9-s + (−0.756 + 1.13i)10-s + (1.88 − 0.375i)11-s + (−0.185 − 0.277i)12-s + (−0.0385 − 0.0385i)13-s + (0.494 + 0.0984i)14-s + (−0.651 − 1.57i)15-s + 1.18i·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.771789 - 0.888484i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.771789 - 0.888484i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + (2.05 - 0.852i)T + (2.82 - 2.82i)T^{2} \) |
| 3 | \( 1 + (-0.814 + 4.09i)T + (-8.31 - 3.44i)T^{2} \) |
| 5 | \( 1 + (-5.08 + 3.39i)T + (9.56 - 23.0i)T^{2} \) |
| 7 | \( 1 + (2.63 + 1.76i)T + (18.7 + 45.2i)T^{2} \) |
| 11 | \( 1 + (-20.7 + 4.12i)T + (111. - 46.3i)T^{2} \) |
| 13 | \( 1 + (0.501 + 0.501i)T + 169iT^{2} \) |
| 19 | \( 1 + (-2.46 + 1.02i)T + (255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-2.89 - 14.5i)T + (-488. + 202. i)T^{2} \) |
| 29 | \( 1 + (18.2 + 27.2i)T + (-321. + 776. i)T^{2} \) |
| 31 | \( 1 + (32.9 + 6.56i)T + (887. + 367. i)T^{2} \) |
| 37 | \( 1 + (-2.73 + 13.7i)T + (-1.26e3 - 523. i)T^{2} \) |
| 41 | \( 1 + (0.0209 + 0.0139i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-34.9 - 14.4i)T + (1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (28.9 + 28.9i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-12.1 + 5.02i)T + (1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (1.70 - 4.12i)T + (-2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (12.9 - 19.3i)T + (-1.42e3 - 3.43e3i)T^{2} \) |
| 67 | \( 1 - 49.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (22.5 - 113. i)T + (-4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-98.9 + 66.1i)T + (2.03e3 - 4.92e3i)T^{2} \) |
| 79 | \( 1 + (67.2 - 13.3i)T + (5.76e3 - 2.38e3i)T^{2} \) |
| 83 | \( 1 + (15.8 + 38.3i)T + (-4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (102. - 102. i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (-38.6 - 57.8i)T + (-3.60e3 + 8.69e3i)T^{2} \) |
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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
−11.41090447544796342410152059911, −9.817257752237449802073801706988, −9.253772239143370381486389188602, −8.528094789032905430236053283216, −7.43651477274341728066107966287, −6.69353952251045948381679241484, −5.84455207198219804844154763427, −3.86083841752421540064097710328, −1.77744529855682304620002326923, −0.877992385017321315060483185364,
1.74394042929999940597407821759, 3.21167009615613858768588158169, 4.52766801069545475037903866695, 5.84783564699363089351532100724, 6.99702463767956719449263222767, 8.706290537670900188526025104257, 9.416249126971638087821461565626, 9.672899576811276583658746427637, 10.60700395944039582739598111280, 11.24546187651748866434611968501
