| L(s) = 1 | + (1.24 − 0.664i)2-s + (0.443 − 2.93i)3-s + (1.11 − 1.65i)4-s + (−3.67 + 1.44i)5-s + (−1.39 − 3.96i)6-s + (2.36 + 1.18i)7-s + (0.292 − 2.81i)8-s + (−5.57 − 1.72i)9-s + (−3.63 + 4.24i)10-s + (−1.30 − 4.24i)11-s + (−4.38 − 4.01i)12-s + (−1.69 − 0.387i)13-s + (3.74 − 0.0859i)14-s + (2.61 + 11.4i)15-s + (−1.50 − 3.70i)16-s + (0.673 + 0.459i)17-s + ⋯ |
| L(s) = 1 | + (0.882 − 0.469i)2-s + (0.255 − 1.69i)3-s + (0.558 − 0.829i)4-s + (−1.64 + 0.645i)5-s + (−0.571 − 1.61i)6-s + (0.893 + 0.449i)7-s + (0.103 − 0.994i)8-s + (−1.85 − 0.573i)9-s + (−1.14 + 1.34i)10-s + (−0.394 − 1.27i)11-s + (−1.26 − 1.16i)12-s + (−0.471 − 0.107i)13-s + (0.999 − 0.0229i)14-s + (0.674 + 2.95i)15-s + (−0.375 − 0.926i)16-s + (0.163 + 0.111i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.460637 - 1.83366i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.460637 - 1.83366i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.24 + 0.664i)T \) |
| 7 | \( 1 + (-2.36 - 1.18i)T \) |
| good | 3 | \( 1 + (-0.443 + 2.93i)T + (-2.86 - 0.884i)T^{2} \) |
| 5 | \( 1 + (3.67 - 1.44i)T + (3.66 - 3.40i)T^{2} \) |
| 11 | \( 1 + (1.30 + 4.24i)T + (-9.08 + 6.19i)T^{2} \) |
| 13 | \( 1 + (1.69 + 0.387i)T + (11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (-0.673 - 0.459i)T + (6.21 + 15.8i)T^{2} \) |
| 19 | \( 1 + (-5.68 - 3.27i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.85 + 2.63i)T + (8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (-2.32 - 4.82i)T + (-18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (0.0831 + 0.144i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.25 - 0.468i)T + (36.5 + 5.51i)T^{2} \) |
| 41 | \( 1 + (1.52 - 1.91i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (-1.43 + 1.14i)T + (9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (5.80 + 5.38i)T + (3.51 + 46.8i)T^{2} \) |
| 53 | \( 1 + (-1.35 + 0.101i)T + (52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (-5.50 - 2.16i)T + (43.2 + 40.1i)T^{2} \) |
| 61 | \( 1 + (7.34 + 0.550i)T + (60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (1.43 - 0.830i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.00 + 0.966i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (2.69 - 2.50i)T + (5.45 - 72.7i)T^{2} \) |
| 79 | \( 1 + (1.37 - 2.38i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.52 + 1.26i)T + (74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (3.20 + 0.988i)T + (73.5 + 50.1i)T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−11.50013945976158093441123878095, −10.60848930865925904664612644212, −8.635588397623192089512975766023, −7.82108746209701757297778030323, −7.26878500355889044717217061944, −6.20138385530871650627083605269, −5.05908291801918250877747568379, −3.42827877753785447934738161391, −2.67558910048250118579806127205, −0.992511108754166746739538710771,
3.03923147505544451832890828907, 4.14200362378739254527813456777, 4.70325491565488377985906536649, 5.14693676696622659699488954855, 7.36029830661618769377949441354, 7.78717501949961231325682034583, 8.830266521476033128153307514631, 9.867029516971827826802732144545, 11.14112444411931055634641806698, 11.53700249316631375898038598949
