| L(s) = 1 | + (3.92 + 12.0i)2-s + (21.8 + 15.8i)3-s + (−27.1 + 19.7i)4-s + (−17.6 + 54.3i)5-s + (−106. + 326. i)6-s + (−962. + 699. i)7-s + (971. + 705. i)8-s + (225. + 693. i)9-s − 726.·10-s + (−1.59e3 + 4.11e3i)11-s − 906.·12-s + (−158. − 488. i)13-s + (−1.22e4 − 8.88e3i)14-s + (−1.24e3 + 906. i)15-s + (−6.04e3 + 1.85e4i)16-s + (5.67e3 − 1.74e4i)17-s + ⋯ |
| L(s) = 1 | + (0.347 + 1.06i)2-s + (0.467 + 0.339i)3-s + (−0.212 + 0.154i)4-s + (−0.0631 + 0.194i)5-s + (−0.200 + 0.616i)6-s + (−1.06 + 0.770i)7-s + (0.670 + 0.487i)8-s + (0.103 + 0.317i)9-s − 0.229·10-s + (−0.360 + 0.932i)11-s − 0.151·12-s + (−0.0200 − 0.0617i)13-s + (−1.19 − 0.865i)14-s + (−0.0954 + 0.0693i)15-s + (−0.368 + 1.13i)16-s + (0.279 − 0.861i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 - 0.402i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.915 - 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.441778 + 2.09968i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.441778 + 2.09968i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-21.8 - 15.8i)T \) |
| 11 | \( 1 + (1.59e3 - 4.11e3i)T \) |
| good | 2 | \( 1 + (-3.92 - 12.0i)T + (-103. + 75.2i)T^{2} \) |
| 5 | \( 1 + (17.6 - 54.3i)T + (-6.32e4 - 4.59e4i)T^{2} \) |
| 7 | \( 1 + (962. - 699. i)T + (2.54e5 - 7.83e5i)T^{2} \) |
| 13 | \( 1 + (158. + 488. i)T + (-5.07e7 + 3.68e7i)T^{2} \) |
| 17 | \( 1 + (-5.67e3 + 1.74e4i)T + (-3.31e8 - 2.41e8i)T^{2} \) |
| 19 | \( 1 + (1.12e4 + 8.20e3i)T + (2.76e8 + 8.50e8i)T^{2} \) |
| 23 | \( 1 + 2.05e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (-1.67e5 + 1.21e5i)T + (5.33e9 - 1.64e10i)T^{2} \) |
| 31 | \( 1 + (-7.84e4 - 2.41e5i)T + (-2.22e10 + 1.61e10i)T^{2} \) |
| 37 | \( 1 + (-2.35e5 + 1.70e5i)T + (2.93e10 - 9.02e10i)T^{2} \) |
| 41 | \( 1 + (-3.82e3 - 2.78e3i)T + (6.01e10 + 1.85e11i)T^{2} \) |
| 43 | \( 1 - 7.64e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (3.25e5 + 2.36e5i)T + (1.56e11 + 4.81e11i)T^{2} \) |
| 53 | \( 1 + (-2.06e5 - 6.36e5i)T + (-9.50e11 + 6.90e11i)T^{2} \) |
| 59 | \( 1 + (1.53e6 - 1.11e6i)T + (7.69e11 - 2.36e12i)T^{2} \) |
| 61 | \( 1 + (2.46e5 - 7.57e5i)T + (-2.54e12 - 1.84e12i)T^{2} \) |
| 67 | \( 1 + 6.08e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + (-1.80e6 + 5.54e6i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 + (-1.60e6 + 1.16e6i)T + (3.41e12 - 1.05e13i)T^{2} \) |
| 79 | \( 1 + (1.50e6 + 4.61e6i)T + (-1.55e13 + 1.12e13i)T^{2} \) |
| 83 | \( 1 + (-2.09e6 + 6.44e6i)T + (-2.19e13 - 1.59e13i)T^{2} \) |
| 89 | \( 1 - 6.53e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-2.51e6 - 7.73e6i)T + (-6.53e13 + 4.74e13i)T^{2} \) |
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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
−15.63663488438427588191671207537, −14.81546678431554449569060688426, −13.63503254603115401687112592061, −12.31446028956326951290005242543, −10.43526882087277666322658147737, −9.119454993362083264002697199430, −7.54861126312137856012217367149, −6.31176963074602693551782120666, −4.80265504181799799788754455227, −2.67699655825456463294952317913,
0.867300744612508579212087166592, 2.80094923104168126277049724697, 3.99819112875221658521851836216, 6.50684275935577141821927561175, 8.112380893856744748043468419729, 9.873013166073640840132232118669, 10.90342512189180070461349840863, 12.43127839150637795538368161584, 13.13907262212031108832891038977, 14.13593970422044581194878186516
