| L(s) = 1 | + (3.09 − 2.87i)2-s + (−6.91 + 1.04i)3-s + (0.734 − 9.80i)4-s + (7.54 − 19.2i)5-s + (−18.4 + 23.1i)6-s + (−18.1 − 3.57i)7-s + (−4.82 − 6.05i)8-s + (20.9 − 6.47i)9-s + (−31.8 − 81.1i)10-s + (61.5 + 18.9i)11-s + (5.14 + 68.6i)12-s + (−0.576 − 2.52i)13-s + (−66.5 + 41.1i)14-s + (−32.1 + 140. i)15-s + (45.4 + 6.85i)16-s + (13.6 − 9.27i)17-s + ⋯ |
| L(s) = 1 | + (1.09 − 1.01i)2-s + (−1.33 + 0.200i)3-s + (0.0918 − 1.22i)4-s + (0.674 − 1.71i)5-s + (−1.25 + 1.57i)6-s + (−0.981 − 0.192i)7-s + (−0.213 − 0.267i)8-s + (0.777 − 0.239i)9-s + (−1.00 − 2.56i)10-s + (1.68 + 0.520i)11-s + (0.123 + 1.65i)12-s + (−0.0123 − 0.0538i)13-s + (−1.26 + 0.785i)14-s + (−0.553 + 2.42i)15-s + (0.710 + 0.107i)16-s + (0.194 − 0.132i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.539 + 0.841i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.539 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.783151 - 1.43191i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.783151 - 1.43191i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (18.1 + 3.57i)T \) |
| good | 2 | \( 1 + (-3.09 + 2.87i)T + (0.597 - 7.97i)T^{2} \) |
| 3 | \( 1 + (6.91 - 1.04i)T + (25.8 - 7.95i)T^{2} \) |
| 5 | \( 1 + (-7.54 + 19.2i)T + (-91.6 - 85.0i)T^{2} \) |
| 11 | \( 1 + (-61.5 - 18.9i)T + (1.09e3 + 749. i)T^{2} \) |
| 13 | \( 1 + (0.576 + 2.52i)T + (-1.97e3 + 953. i)T^{2} \) |
| 17 | \( 1 + (-13.6 + 9.27i)T + (1.79e3 - 4.57e3i)T^{2} \) |
| 19 | \( 1 + (-9.60 - 16.6i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (11.6 + 7.94i)T + (4.44e3 + 1.13e4i)T^{2} \) |
| 29 | \( 1 + (-25.9 - 12.5i)T + (1.52e4 + 1.90e4i)T^{2} \) |
| 31 | \( 1 + (97.9 - 169. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (10.7 + 143. i)T + (-5.00e4 + 7.54e3i)T^{2} \) |
| 41 | \( 1 + (-59.5 - 74.6i)T + (-1.53e4 + 6.71e4i)T^{2} \) |
| 43 | \( 1 + (153. - 192. i)T + (-1.76e4 - 7.75e4i)T^{2} \) |
| 47 | \( 1 + (44.1 - 40.9i)T + (7.75e3 - 1.03e5i)T^{2} \) |
| 53 | \( 1 + (3.90 - 52.0i)T + (-1.47e5 - 2.21e4i)T^{2} \) |
| 59 | \( 1 + (-25.5 - 65.1i)T + (-1.50e5 + 1.39e5i)T^{2} \) |
| 61 | \( 1 + (-30.0 - 401. i)T + (-2.24e5 + 3.38e4i)T^{2} \) |
| 67 | \( 1 + (235. - 407. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-452. + 217. i)T + (2.23e5 - 2.79e5i)T^{2} \) |
| 73 | \( 1 + (-270. - 251. i)T + (2.90e4 + 3.87e5i)T^{2} \) |
| 79 | \( 1 + (182. + 316. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-157. + 687. i)T + (-5.15e5 - 2.48e5i)T^{2} \) |
| 89 | \( 1 + (-96.5 + 29.7i)T + (5.82e5 - 3.97e5i)T^{2} \) |
| 97 | \( 1 - 842.T + 9.12e5T^{2} \) |
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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
−14.25017943358722044881568530536, −13.05458068890611812190915160966, −12.34164328401794744747150315989, −11.74744572626494715831766401244, −10.23378025324024182771886725680, −9.192086911690256986323374564926, −6.26994788332949886002253669410, −5.20438526275426273486175868924, −4.12516356095595111689006246887, −1.22310679727384655307898739915,
3.55509103682028065490573077295, 5.74267187647683554823255836073, 6.42127696070596454710630606766, 6.94692992909006061290851265715, 9.774769647120688329446985747173, 11.08768175412721146576625692926, 12.15892372139718031080604703586, 13.52545860628656349949880492808, 14.37626720603429030628430302375, 15.33592159715389918585135205382
