L(s) = 1 | + (−0.654 + 0.755i)2-s + (0.200 + 0.439i)3-s + (−0.142 − 0.989i)4-s + (0.959 − 0.281i)5-s + (−0.463 − 0.136i)6-s + (−1.81 + 2.09i)7-s + (0.841 + 0.540i)8-s + (1.81 − 2.09i)9-s + (−0.415 + 0.909i)10-s + (2.88 − 0.847i)11-s + (0.406 − 0.261i)12-s + (2.41 − 1.54i)13-s + (−0.393 − 2.73i)14-s + (0.316 + 0.365i)15-s + (−0.959 + 0.281i)16-s + (−0.0159 + 0.111i)17-s + ⋯ |
L(s) = 1 | + (−0.463 + 0.534i)2-s + (0.115 + 0.253i)3-s + (−0.0711 − 0.494i)4-s + (0.429 − 0.125i)5-s + (−0.189 − 0.0555i)6-s + (−0.685 + 0.790i)7-s + (0.297 + 0.191i)8-s + (0.603 − 0.696i)9-s + (−0.131 + 0.287i)10-s + (0.870 − 0.255i)11-s + (0.117 − 0.0753i)12-s + (0.668 − 0.429i)13-s + (−0.105 − 0.732i)14-s + (0.0816 + 0.0942i)15-s + (−0.239 + 0.0704i)16-s + (−0.00387 + 0.0269i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.28903 + 0.479487i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28903 + 0.479487i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.654 - 0.755i)T \) |
| 5 | \( 1 + (-0.959 + 0.281i)T \) |
| 67 | \( 1 + (7.99 - 1.75i)T \) |
good | 3 | \( 1 + (-0.200 - 0.439i)T + (-1.96 + 2.26i)T^{2} \) |
| 7 | \( 1 + (1.81 - 2.09i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-2.88 + 0.847i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-2.41 + 1.54i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (0.0159 - 0.111i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (2.93 + 3.38i)T + (-2.70 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-2.21 - 4.85i)T + (-15.0 + 17.3i)T^{2} \) |
| 29 | \( 1 - 2.67T + 29T^{2} \) |
| 31 | \( 1 + (-5.13 - 3.30i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 - 3.50T + 37T^{2} \) |
| 41 | \( 1 + (-0.446 + 3.10i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (0.448 - 3.11i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + (-1.08 - 2.37i)T + (-30.7 + 35.5i)T^{2} \) |
| 53 | \( 1 + (0.519 + 3.61i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-1.04 - 0.669i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-3.72 - 1.09i)T + (51.3 + 32.9i)T^{2} \) |
| 71 | \( 1 + (-0.510 - 3.54i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-13.2 - 3.88i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-3.81 + 2.44i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (0.483 - 0.141i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (4.76 - 10.4i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + 7.65T + 97T^{2} \) |
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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
−10.33008871254246030963283240508, −9.426171527936154489270080216909, −9.085632481712201594758299623774, −8.221766848134387952172538865117, −6.80476509414649616475971138743, −6.35390090864171060595553130403, −5.40310976021168818600666834584, −4.11827769913387317247407292014, −2.89897676940726264867693797174, −1.17049629920449784788962672072,
1.16783037057040043150875336146, 2.36012672856867949825291104569, 3.75667308140023456351235641921, 4.56700338013044041688098810258, 6.31328646557960545693566793023, 6.83340251840794100835912491428, 7.891027743612434687596005219354, 8.787134023116727587197003304964, 9.746786066232230196658327101700, 10.33165792841000364772807398183