L(s) = 1 | + (−0.206 − 0.636i)2-s + (−2.54 − 1.84i)3-s + (1.25 − 0.911i)4-s + (0.662 − 2.03i)5-s + (−0.649 + 1.99i)6-s + (0.809 − 0.587i)7-s + (−1.92 − 1.39i)8-s + (2.11 + 6.52i)9-s − 1.43·10-s − 4.87·12-s + (−0.781 − 2.40i)13-s + (−0.541 − 0.393i)14-s + (−5.44 + 3.95i)15-s + (0.466 − 1.43i)16-s + (0.553 − 1.70i)17-s + (3.71 − 2.69i)18-s + ⋯ |
L(s) = 1 | + (−0.146 − 0.450i)2-s + (−1.46 − 1.06i)3-s + (0.627 − 0.455i)4-s + (0.296 − 0.911i)5-s + (−0.265 + 0.816i)6-s + (0.305 − 0.222i)7-s + (−0.680 − 0.494i)8-s + (0.706 + 2.17i)9-s − 0.454·10-s − 1.40·12-s + (−0.216 − 0.666i)13-s + (−0.144 − 0.105i)14-s + (−1.40 + 1.02i)15-s + (0.116 − 0.358i)16-s + (0.134 − 0.413i)17-s + (0.875 − 0.636i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.349335 + 0.704652i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.349335 + 0.704652i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.206 + 0.636i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (2.54 + 1.84i)T + (0.927 + 2.85i)T^{2} \) |
| 5 | \( 1 + (-0.662 + 2.03i)T + (-4.04 - 2.93i)T^{2} \) |
| 13 | \( 1 + (0.781 + 2.40i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.553 + 1.70i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (5.44 + 3.95i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + 3.16T + 23T^{2} \) |
| 29 | \( 1 + (0.747 - 0.543i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.927 - 2.85i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.21 + 0.885i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-4.49 - 3.26i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 8.42T + 43T^{2} \) |
| 47 | \( 1 + (3.55 + 2.58i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-0.206 - 0.634i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-0.298 + 0.216i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (1.54 - 4.76i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + 0.902T + 67T^{2} \) |
| 71 | \( 1 + (4.59 - 14.1i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-6.50 + 4.72i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (1.25 + 3.85i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-1.25 + 3.85i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 8.30T + 89T^{2} \) |
| 97 | \( 1 + (-2.63 - 8.09i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−9.990122263970361523284281973948, −8.844172745897177471335193304214, −7.69969727384119465299430148279, −6.91678632058389129805945573667, −6.08443332862183391589408752964, −5.41225654735158099125476733254, −4.57600590689308087711049931665, −2.46870758271257967224942468193, −1.39540835439132277789270833762, −0.48846211109336684892563944342,
2.20200036900524568061805870360, 3.66621477045785173398896372685, 4.55578768640356797930456126700, 5.94329320044658662253284641862, 6.12193510070244420401988551606, 7.01280437585670462729135094145, 8.089908511615901350788716103617, 9.226430777764219417192272282323, 10.15710070223360516474374964032, 10.80321597797710410504328113127