| L(s) = 1 | + (−0.610 + 0.163i)2-s + (0.608 − 0.793i)3-s + (−1.38 + 0.800i)4-s + (−0.0694 + 0.527i)5-s + (−0.241 + 0.583i)6-s + (−2.47 + 0.924i)7-s + (1.60 − 1.60i)8-s + (−0.258 − 0.965i)9-s + (−0.0438 − 0.333i)10-s + (−2.95 + 0.388i)11-s + (−0.208 + 1.58i)12-s − 7.01i·13-s + (1.36 − 0.969i)14-s + (0.376 + 0.376i)15-s + (0.882 − 1.52i)16-s + (−3.98 + 1.06i)17-s + ⋯ |
| L(s) = 1 | + (−0.431 + 0.115i)2-s + (0.351 − 0.458i)3-s + (−0.693 + 0.400i)4-s + (−0.0310 + 0.236i)5-s + (−0.0986 + 0.238i)6-s + (−0.936 + 0.349i)7-s + (0.568 − 0.568i)8-s + (−0.0862 − 0.321i)9-s + (−0.0138 − 0.105i)10-s + (−0.890 + 0.117i)11-s + (−0.0603 + 0.458i)12-s − 1.94i·13-s + (0.363 − 0.259i)14-s + (0.0971 + 0.0971i)15-s + (0.220 − 0.382i)16-s + (−0.965 + 0.259i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 + 0.457i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0477194 - 0.196845i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0477194 - 0.196845i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.608 + 0.793i)T \) |
| 7 | \( 1 + (2.47 - 0.924i)T \) |
| 17 | \( 1 + (3.98 - 1.06i)T \) |
| good | 2 | \( 1 + (0.610 - 0.163i)T + (1.73 - i)T^{2} \) |
| 5 | \( 1 + (0.0694 - 0.527i)T + (-4.82 - 1.29i)T^{2} \) |
| 11 | \( 1 + (2.95 - 0.388i)T + (10.6 - 2.84i)T^{2} \) |
| 13 | \( 1 + 7.01iT - 13T^{2} \) |
| 19 | \( 1 + (3.45 - 0.925i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (3.61 + 4.71i)T + (-5.95 + 22.2i)T^{2} \) |
| 29 | \( 1 + (6.59 - 2.73i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + (2.37 - 3.09i)T + (-8.02 - 29.9i)T^{2} \) |
| 37 | \( 1 + (-6.56 - 0.864i)T + (35.7 + 9.57i)T^{2} \) |
| 41 | \( 1 + (6.09 + 2.52i)T + (28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (-0.374 + 0.374i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.462 + 0.267i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.445 - 1.66i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-1.39 - 0.372i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-6.43 + 4.94i)T + (15.7 - 58.9i)T^{2} \) |
| 67 | \( 1 + (2.02 + 3.50i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.65 - 8.81i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (11.0 + 8.45i)T + (18.8 + 70.5i)T^{2} \) |
| 79 | \( 1 + (1.55 + 2.03i)T + (-20.4 + 76.3i)T^{2} \) |
| 83 | \( 1 + (-6.29 - 6.29i)T + 83iT^{2} \) |
| 89 | \( 1 + (-10.9 - 6.32i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.00 + 2.07i)T + (68.5 - 68.5i)T^{2} \) |
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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
−10.69942447753265048343273164929, −10.14738461677737405755318436628, −8.998085935527843112907199719895, −8.306301868242576560869376328235, −7.50425211369917518452330081833, −6.41342961579300374480574239165, −5.19566703585155647342813033742, −3.65718461198737926410661477379, −2.61841084086847528831460486827, −0.14656183613551381160741983492,
2.15024796232839265656503882152, 3.93480217325229037425951891846, 4.68421145350216502295630149964, 6.02784006123433392021377064132, 7.23597726239333504207894108301, 8.465659235239767015907474473922, 9.266868177347009674297916665790, 9.735509422595711090036531716947, 10.73392251727117912152212105692, 11.55711800396334960772698584369
