| L(s) = 1 | + (1.84 + 1.06i)2-s + (−15.5 − 1.62i)3-s + (−13.7 − 23.7i)4-s + (−47.5 + 82.3i)5-s + (−26.8 − 19.5i)6-s + (−95.0 − 88.1i)7-s − 126. i·8-s + (237. + 50.5i)9-s + (−175. + 101. i)10-s + (−113. + 65.3i)11-s + (174. + 390. i)12-s + 14.6i·13-s + (−81.5 − 264. i)14-s + (870. − 1.19e3i)15-s + (−304. + 526. i)16-s + (320. + 554. i)17-s + ⋯ |
| L(s) = 1 | + (0.326 + 0.188i)2-s + (−0.994 − 0.104i)3-s + (−0.428 − 0.742i)4-s + (−0.850 + 1.47i)5-s + (−0.305 − 0.221i)6-s + (−0.733 − 0.680i)7-s − 0.700i·8-s + (0.978 + 0.207i)9-s + (−0.555 + 0.320i)10-s + (−0.282 + 0.162i)11-s + (0.348 + 0.783i)12-s + 0.0240i·13-s + (−0.111 − 0.360i)14-s + (0.999 − 1.37i)15-s + (−0.296 + 0.514i)16-s + (0.268 + 0.465i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0678i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.000709045 - 0.0208672i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000709045 - 0.0208672i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (15.5 + 1.62i)T \) |
| 7 | \( 1 + (95.0 + 88.1i)T \) |
| good | 2 | \( 1 + (-1.84 - 1.06i)T + (16 + 27.7i)T^{2} \) |
| 5 | \( 1 + (47.5 - 82.3i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (113. - 65.3i)T + (8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 - 14.6iT - 3.71e5T^{2} \) |
| 17 | \( 1 + (-320. - 554. i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (469. + 271. i)T + (1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (3.54e3 + 2.04e3i)T + (3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 - 3.59e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + (2.51e3 - 1.45e3i)T + (1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-2.17e3 + 3.76e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + 1.09e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.05e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-4.40e3 + 7.63e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (6.04e3 - 3.48e3i)T + (2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-2.00e4 - 3.46e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (2.87e4 + 1.66e4i)T + (4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (3.21e3 + 5.57e3i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 3.24e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (1.10e4 - 6.36e3i)T + (1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (4.29e3 - 7.44e3i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 3.16e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (1.59e4 - 2.76e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + 1.43e5iT - 8.58e9T^{2} \) |
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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
−17.99877588740173263355617083277, −16.27706231626451006167510595871, −15.23853226701236088950867867767, −14.03552695213780715046897557152, −12.51197842213778419241338670294, −10.82677737901514295949500112466, −10.18448182331777039079418299667, −7.21997425428852672370029352901, −6.15791476010781549343366289937, −4.04590542000420282938308236956,
0.01531368913618527638930941540, 4.08183687216712186855829240551, 5.48895097957535099920452816865, 7.951611765431253602728538118396, 9.410144756824063259619815465506, 11.70114302236947963059523131787, 12.34300365920362730756419072573, 13.24411683568748040224860559013, 15.74639901407362230566482179778, 16.38654836093732487249552686652
