| L(s) = 1 | + (0.415 − 0.909i)2-s + (2.84 + 0.835i)3-s + (−0.654 − 0.755i)4-s + (−0.221 + 0.142i)5-s + (1.94 − 2.24i)6-s + (0.142 − 0.989i)7-s + (−0.959 + 0.281i)8-s + (4.87 + 3.13i)9-s + (0.0374 + 0.260i)10-s + (−0.940 − 2.05i)11-s + (−1.23 − 2.69i)12-s + (0.349 + 2.43i)13-s + (−0.841 − 0.540i)14-s + (−0.748 + 0.219i)15-s + (−0.142 + 0.989i)16-s + (−0.0868 + 0.100i)17-s + ⋯ |
| L(s) = 1 | + (0.293 − 0.643i)2-s + (1.64 + 0.482i)3-s + (−0.327 − 0.377i)4-s + (−0.0989 + 0.0636i)5-s + (0.792 − 0.914i)6-s + (0.0537 − 0.374i)7-s + (−0.339 + 0.0996i)8-s + (1.62 + 1.04i)9-s + (0.0118 + 0.0823i)10-s + (−0.283 − 0.620i)11-s + (−0.355 − 0.778i)12-s + (0.0970 + 0.674i)13-s + (−0.224 − 0.144i)14-s + (−0.193 + 0.0567i)15-s + (−0.0355 + 0.247i)16-s + (−0.0210 + 0.0243i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 + 0.527i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.23236 - 0.636122i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.23236 - 0.636122i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 + 0.909i)T \) |
| 7 | \( 1 + (-0.142 + 0.989i)T \) |
| 23 | \( 1 + (-4.01 - 2.62i)T \) |
| good | 3 | \( 1 + (-2.84 - 0.835i)T + (2.52 + 1.62i)T^{2} \) |
| 5 | \( 1 + (0.221 - 0.142i)T + (2.07 - 4.54i)T^{2} \) |
| 11 | \( 1 + (0.940 + 2.05i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.349 - 2.43i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (0.0868 - 0.100i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (3.38 + 3.90i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (0.566 - 0.653i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (8.27 - 2.42i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (4.71 + 3.02i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-1.66 + 1.07i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-8.88 - 2.61i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + (-1.21 + 8.44i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (1.14 + 7.93i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (5.34 - 1.56i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-1.08 + 2.38i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (0.253 - 0.555i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-4.66 - 5.38i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.391 - 2.72i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-13.1 - 8.46i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-1.57 - 0.461i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (9.34 - 6.00i)T + (40.2 - 88.2i)T^{2} \) |
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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
−11.24635814350421018594097063871, −10.70329215387794008801245615934, −9.428096273522422867042445941829, −9.036040843583054870364117764490, −7.996670436846419565155136111693, −6.92657952823885803796323916853, −5.13776707177570780301040451203, −3.94796149122903784784660114134, −3.21269514748209351760918537570, −1.94775133736400807881961919546,
2.11608915794431859372545183881, 3.30570359534632141402725150663, 4.47560210034885880362760672639, 5.96297824354546111129857707779, 7.19239998012010241600318876075, 7.918792469570129709851556365598, 8.624601412733459880935153321023, 9.443680723748427082794163450743, 10.56465200445105948266092323071, 12.28054374008661419599473496978
