| L(s) = 1 | + (0.959 + 0.281i)2-s + (−0.442 + 3.07i)3-s + (0.841 + 0.540i)4-s + (0.415 + 0.909i)5-s + (−1.29 + 2.83i)6-s + (0.952 + 0.279i)7-s + (0.654 + 0.755i)8-s + (−6.40 − 1.88i)9-s + (0.142 + 0.989i)10-s + (0.513 + 1.12i)11-s + (−2.03 + 2.35i)12-s + (−1.87 + 2.16i)13-s + (0.834 + 0.536i)14-s + (−2.98 + 0.876i)15-s + (0.415 + 0.909i)16-s + (1.78 − 1.14i)17-s + ⋯ |
| L(s) = 1 | + (0.678 + 0.199i)2-s + (−0.255 + 1.77i)3-s + (0.420 + 0.270i)4-s + (0.185 + 0.406i)5-s + (−0.527 + 1.15i)6-s + (0.359 + 0.105i)7-s + (0.231 + 0.267i)8-s + (−2.13 − 0.627i)9-s + (0.0450 + 0.313i)10-s + (0.154 + 0.339i)11-s + (−0.588 + 0.678i)12-s + (−0.520 + 0.600i)13-s + (0.223 + 0.143i)14-s + (−0.770 + 0.226i)15-s + (0.103 + 0.227i)16-s + (0.433 − 0.278i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.384676 + 1.97477i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.384676 + 1.97477i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.959 - 0.281i)T \) |
| 5 | \( 1 + (-0.415 - 0.909i)T \) |
| 67 | \( 1 + (8.07 + 1.34i)T \) |
| good | 3 | \( 1 + (0.442 - 3.07i)T + (-2.87 - 0.845i)T^{2} \) |
| 7 | \( 1 + (-0.952 - 0.279i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (-0.513 - 1.12i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (1.87 - 2.16i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.78 + 1.14i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-0.440 + 0.129i)T + (15.9 - 10.2i)T^{2} \) |
| 23 | \( 1 + (-0.346 + 2.41i)T + (-22.0 - 6.47i)T^{2} \) |
| 29 | \( 1 - 1.15T + 29T^{2} \) |
| 31 | \( 1 + (0.121 + 0.139i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 - 6.05T + 37T^{2} \) |
| 41 | \( 1 + (7.00 - 4.50i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-6.91 + 4.44i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (-1.16 + 8.07i)T + (-45.0 - 13.2i)T^{2} \) |
| 53 | \( 1 + (-4.98 - 3.20i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (0.771 + 0.890i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (4.23 - 9.26i)T + (-39.9 - 46.1i)T^{2} \) |
| 71 | \( 1 + (-8.89 - 5.71i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (0.565 - 1.23i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-3.86 + 4.46i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-5.25 - 11.5i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-2.24 - 15.6i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 - 9.56T + 97T^{2} \) |
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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
−10.84692223735503228020058596241, −10.08856151697166222855392514645, −9.415065888370195988259334501483, −8.452849065077329473242788392403, −7.19521274496684757040312389635, −6.11720983114283085617581314396, −5.18361092674852696288983863030, −4.52768296983768986024963804923, −3.64380501122735601452478384767, −2.51510420891607363903064211127,
0.920534454281706595713086130610, 1.99982933265776892506688021290, 3.18022144926755051849094017742, 4.82567173250107740005241751210, 5.76384177603603968309836458226, 6.39556890599647286490997826246, 7.54245096121984849327516213142, 7.949055408420504015037176562929, 9.131301463498246520475349599694, 10.42760848257817759920814490691
