| L(s) = 1 | + (−4 + 6.92i)5-s + (−20 − 34.6i)11-s − 12·13-s + (−29 − 50.2i)17-s + (−13 + 22.5i)19-s + (−32 + 55.4i)23-s + (30.5 + 52.8i)25-s + 62·29-s + (−126 − 218. i)31-s + (−13 + 22.5i)37-s − 6·41-s + 416·43-s + (−198 + 342. i)47-s + (−225 − 389. i)53-s + 320·55-s + ⋯ |
| L(s) = 1 | + (−0.357 + 0.619i)5-s + (−0.548 − 0.949i)11-s − 0.256·13-s + (−0.413 − 0.716i)17-s + (−0.156 + 0.271i)19-s + (−0.290 + 0.502i)23-s + (0.244 + 0.422i)25-s + 0.397·29-s + (−0.730 − 1.26i)31-s + (−0.0577 + 0.100i)37-s − 0.0228·41-s + 1.47·43-s + (−0.614 + 1.06i)47-s + (−0.583 − 1.01i)53-s + 0.784·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.316514793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.316514793\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (4 - 6.92i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (20 + 34.6i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 12T + 2.19e3T^{2} \) |
| 17 | \( 1 + (29 + 50.2i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (13 - 22.5i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (32 - 55.4i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 62T + 2.43e4T^{2} \) |
| 31 | \( 1 + (126 + 218. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (13 - 22.5i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 416T + 7.95e4T^{2} \) |
| 47 | \( 1 + (198 - 342. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (225 + 389. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-137 - 237. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-288 + 498. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-238 - 412. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 448T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-79 - 136. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-468 + 810. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 530T + 5.71e5T^{2} \) |
| 89 | \( 1 + (195 - 337. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 214T + 9.12e5T^{2} \) |
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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
−9.105236130410637616120455831463, −8.098082506055592239600883389068, −7.56403956505456797244465115002, −6.70060123993165709978008939452, −5.85540061850950779536574930553, −5.03051243112049013496082168126, −3.92430849165303129349854604060, −3.10214587634098671619135994730, −2.21778901966792836883252487132, −0.69162672813409337139823330084,
0.40899398155359271184388571442, 1.72746003118927468358898249699, 2.70212319534457241403283708049, 3.99438439144310755964852714049, 4.66937361502452860721835247861, 5.41894259085668671644672603423, 6.53547911758221244498087788488, 7.25059372457522655395115084363, 8.149202634123166378105677685223, 8.708742635729165716083432075491
