| L(s) = 1 | + (−2.64 − 1.70i)3-s + (−2.08 − 0.951i)5-s + (2.90 − 9.89i)7-s + (0.375 + 0.822i)9-s + (6.07 + 5.26i)11-s + (14.4 − 4.25i)13-s + (3.89 + 6.06i)15-s + (−13.1 + 1.89i)17-s + (−26.2 − 3.77i)19-s + (−24.5 + 21.2i)21-s + (−22.1 + 6.20i)23-s + (−12.9 − 14.9i)25-s + (−3.62 + 25.2i)27-s + (0.753 + 5.23i)29-s + (29.7 − 19.1i)31-s + ⋯ |
| L(s) = 1 | + (−0.882 − 0.567i)3-s + (−0.416 − 0.190i)5-s + (0.415 − 1.41i)7-s + (0.0417 + 0.0913i)9-s + (0.552 + 0.478i)11-s + (1.11 − 0.327i)13-s + (0.259 + 0.404i)15-s + (−0.774 + 0.111i)17-s + (−1.38 − 0.198i)19-s + (−1.16 + 1.01i)21-s + (−0.962 + 0.269i)23-s + (−0.517 − 0.597i)25-s + (−0.134 + 0.934i)27-s + (0.0259 + 0.180i)29-s + (0.959 − 0.616i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.129i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0361808 + 0.555741i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0361808 + 0.555741i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (22.1 - 6.20i)T \) |
| good | 3 | \( 1 + (2.64 + 1.70i)T + (3.73 + 8.18i)T^{2} \) |
| 5 | \( 1 + (2.08 + 0.951i)T + (16.3 + 18.8i)T^{2} \) |
| 7 | \( 1 + (-2.90 + 9.89i)T + (-41.2 - 26.4i)T^{2} \) |
| 11 | \( 1 + (-6.07 - 5.26i)T + (17.2 + 119. i)T^{2} \) |
| 13 | \( 1 + (-14.4 + 4.25i)T + (142. - 91.3i)T^{2} \) |
| 17 | \( 1 + (13.1 - 1.89i)T + (277. - 81.4i)T^{2} \) |
| 19 | \( 1 + (26.2 + 3.77i)T + (346. + 101. i)T^{2} \) |
| 29 | \( 1 + (-0.753 - 5.23i)T + (-806. + 236. i)T^{2} \) |
| 31 | \( 1 + (-29.7 + 19.1i)T + (399. - 874. i)T^{2} \) |
| 37 | \( 1 + (32.5 - 14.8i)T + (896. - 1.03e3i)T^{2} \) |
| 41 | \( 1 + (-7.60 + 16.6i)T + (-1.10e3 - 1.27e3i)T^{2} \) |
| 43 | \( 1 + (8.28 - 12.8i)T + (-768. - 1.68e3i)T^{2} \) |
| 47 | \( 1 + 72.6T + 2.20e3T^{2} \) |
| 53 | \( 1 + (25.8 - 88.1i)T + (-2.36e3 - 1.51e3i)T^{2} \) |
| 59 | \( 1 + (-71.1 + 20.8i)T + (2.92e3 - 1.88e3i)T^{2} \) |
| 61 | \( 1 + (15.2 + 23.7i)T + (-1.54e3 + 3.38e3i)T^{2} \) |
| 67 | \( 1 + (-5.29 + 4.58i)T + (638. - 4.44e3i)T^{2} \) |
| 71 | \( 1 + (18.7 + 21.6i)T + (-717. + 4.98e3i)T^{2} \) |
| 73 | \( 1 + (1.65 - 11.5i)T + (-5.11e3 - 1.50e3i)T^{2} \) |
| 79 | \( 1 + (5.91 + 20.1i)T + (-5.25e3 + 3.37e3i)T^{2} \) |
| 83 | \( 1 + (-15.0 + 6.85i)T + (4.51e3 - 5.20e3i)T^{2} \) |
| 89 | \( 1 + (-21.3 + 33.1i)T + (-3.29e3 - 7.20e3i)T^{2} \) |
| 97 | \( 1 + (77.7 + 35.4i)T + (6.16e3 + 7.11e3i)T^{2} \) |
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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
−10.92901382384736758470777381825, −10.08148477128772810572739325173, −8.653222523798736711463357256317, −7.79369155465862230561166157234, −6.71389167520375221255317630241, −6.15648654165954273366426331345, −4.56991325479807609367848172873, −3.84157817929415597977669210947, −1.55643526644017812325113190569, −0.28446721048548324945238383490,
2.06401935271093900894282466748, 3.77201539359640315457514477078, 4.84007499995166203869172278098, 5.91993489435018606390476873849, 6.50881974833246397202213005738, 8.331526684666859359381401171761, 8.682867159960856930624184931206, 9.989559155716286276600064181337, 11.07649436985303502659114034251, 11.46782369455741852902374471912
