| L(s) = 1 | + (−4.5 − 7.79i)3-s + (−15.9 + 27.6i)5-s + (−40.5 + 70.1i)9-s + (−130. − 225. i)11-s − 769.·13-s + 287.·15-s + (−776. − 1.34e3i)17-s + (375. − 649. i)19-s + (−377. + 653. i)23-s + (1.05e3 + 1.82e3i)25-s + 729·27-s + 6.00e3·29-s + (−3.21e3 − 5.55e3i)31-s + (−1.17e3 + 2.03e3i)33-s + (2.38e3 − 4.13e3i)37-s + ⋯ |
| L(s) = 1 | + (−0.288 − 0.499i)3-s + (−0.285 + 0.495i)5-s + (−0.166 + 0.288i)9-s + (−0.325 − 0.562i)11-s − 1.26·13-s + 0.330·15-s + (−0.651 − 1.12i)17-s + (0.238 − 0.412i)19-s + (−0.148 + 0.257i)23-s + (0.336 + 0.582i)25-s + 0.192·27-s + 1.32·29-s + (−0.599 − 1.03i)31-s + (−0.187 + 0.325i)33-s + (0.286 − 0.496i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8138080183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8138080183\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.5 + 7.79i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (15.9 - 27.6i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (130. + 225. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + 769.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (776. + 1.34e3i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-375. + 649. i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (377. - 653. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 - 6.00e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (3.21e3 + 5.55e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-2.38e3 + 4.13e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 - 5.42e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.18e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (8.71e3 - 1.50e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (1.88e4 + 3.26e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-1.10e4 - 1.91e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (4.08e3 - 7.07e3i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (6.50e3 + 1.12e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 1.23e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (-2.18e4 - 3.77e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (3.83e4 - 6.64e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 2.18e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (6.84e4 - 1.18e5i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 9.30e4T + 8.58e9T^{2} \) |
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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
−10.04863039081138182279377917370, −9.231003961595413095069555566307, −8.072530210998777706494762404070, −7.28567970380799604998258816587, −6.64372872747762649061109860804, −5.45420599643476119715017675012, −4.60423421140048572668946027780, −3.11286854645197789354337787934, −2.30236535280910748664620517995, −0.70568049398851539419256735121,
0.26767343996333052369294636123, 1.78406034225996911245922485760, 3.09511539833759666465238796610, 4.46232482733617553076625153097, 4.86497174274568538372822572289, 6.09798772785188995060767031164, 7.09226250658720736496133429294, 8.149229951522144959199136771036, 8.884296405336001080684895419057, 10.01830358340764411590627262832
