L(s) = 1 | + (−1.5 + 2.59i)3-s + (7.5 + 12.9i)5-s + (−17.5 + 6.06i)7-s + (−4.5 − 7.79i)9-s + (−4.5 + 7.79i)11-s − 88·13-s − 45·15-s + (42 − 72.7i)17-s + (52 + 90.0i)19-s + (10.5 − 54.5i)21-s + (−42 − 72.7i)23-s + (−50 + 86.6i)25-s + 27·27-s + 51·29-s + (92.5 − 160. i)31-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (0.670 + 1.16i)5-s + (−0.944 + 0.327i)7-s + (−0.166 − 0.288i)9-s + (−0.123 + 0.213i)11-s − 1.87·13-s − 0.774·15-s + (0.599 − 1.03i)17-s + (0.627 + 1.08i)19-s + (0.109 − 0.566i)21-s + (−0.380 − 0.659i)23-s + (−0.400 + 0.692i)25-s + 0.192·27-s + 0.326·29-s + (0.535 − 0.928i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.2664203726\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2664203726\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 - 2.59i)T \) |
| 7 | \( 1 + (17.5 - 6.06i)T \) |
good | 5 | \( 1 + (-7.5 - 12.9i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (4.5 - 7.79i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 88T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-42 + 72.7i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-52 - 90.0i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (42 + 72.7i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 51T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-92.5 + 160. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (22 + 38.1i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 168T + 6.89e4T^{2} \) |
| 43 | \( 1 + 326T + 7.95e4T^{2} \) |
| 47 | \( 1 + (69 + 119. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (319.5 - 553. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-79.5 + 137. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (361 + 625. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (83 - 143. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 1.08e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + (109 - 188. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (291.5 + 504. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 597T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-519 - 898. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 169T + 9.12e5T^{2} \) |
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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
−11.83391316897089071891655080036, −10.39757026642541822150939783551, −9.944000680502086649920922572583, −9.424134482519009419504758690818, −7.71331329562421992596159268103, −6.78905655987627756655892213963, −5.91382134062253626047663422255, −4.86765044910558577603093834372, −3.24117886410597422240745604571, −2.42983144822964596863136607039,
0.094817040078483643258617881656, 1.47184416633862354614292300231, 2.99703763955553344261495798388, 4.73722222220035555290693618091, 5.52116745542852503046027327156, 6.63525632926821996004962681336, 7.59281829393495495719420288448, 8.744761085666186099424341067458, 9.720259096452749960556992154629, 10.25576065398391225757010550500