L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.653 − 2.43i)5-s + (−1.76 + 3.05i)7-s + (−0.707 + 0.707i)8-s − 2.52·10-s − 2.53·11-s + (−2.84 + 0.762i)13-s + (2.49 + 2.49i)14-s + (0.500 + 0.866i)16-s + (1.73 + 0.463i)17-s + (−3.53 + 0.946i)19-s + (−0.653 + 2.43i)20-s + (−0.656 + 2.44i)22-s + (−3.15 + 3.15i)23-s + ⋯ |
L(s) = 1 | + (0.183 − 0.683i)2-s + (−0.433 − 0.249i)4-s + (−0.292 − 1.09i)5-s + (−0.666 + 1.15i)7-s + (−0.249 + 0.249i)8-s − 0.798·10-s − 0.764·11-s + (−0.788 + 0.211i)13-s + (0.666 + 0.666i)14-s + (0.125 + 0.216i)16-s + (0.419 + 0.112i)17-s + (−0.809 + 0.217i)19-s + (−0.146 + 0.545i)20-s + (−0.139 + 0.521i)22-s + (−0.658 + 0.658i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0542814 + 0.0789563i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0542814 + 0.0789563i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 + 0.965i)T \) |
| 3 | \( 1 \) |
| 37 | \( 1 + (6.08 + 0.00578i)T \) |
good | 5 | \( 1 + (0.653 + 2.43i)T + (-4.33 + 2.5i)T^{2} \) |
| 7 | \( 1 + (1.76 - 3.05i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + 2.53T + 11T^{2} \) |
| 13 | \( 1 + (2.84 - 0.762i)T + (11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (-1.73 - 0.463i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (3.53 - 0.946i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (3.15 - 3.15i)T - 23iT^{2} \) |
| 29 | \( 1 + (0.0393 + 0.0393i)T + 29iT^{2} \) |
| 31 | \( 1 + (3.02 - 3.02i)T - 31iT^{2} \) |
| 41 | \( 1 + (-4.63 + 8.02i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.234 - 0.234i)T + 43iT^{2} \) |
| 47 | \( 1 - 4.39iT - 47T^{2} \) |
| 53 | \( 1 + (-3.09 + 1.78i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (9.18 + 2.46i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-2.55 - 9.52i)T + (-52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (13.5 + 7.80i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.56 + 2.05i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 4.09iT - 73T^{2} \) |
| 79 | \( 1 + (-2.84 + 0.762i)T + (68.4 - 39.5i)T^{2} \) |
| 83 | \( 1 + (7.86 - 4.54i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.78 + 14.1i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (0.841 + 0.841i)T + 97iT^{2} \) |
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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
−10.76129785504004006157738200386, −9.896091021939731938422202659763, −9.070127722000197504055680085796, −8.515080629940045051635058000523, −7.43300867247320274712469319711, −5.93753181383144319698537450689, −5.27114982356675595507044390392, −4.30166319953181783098753243742, −3.03682928393788141046664146063, −1.90115294399461783874900591822,
0.04515174853428826234087614038, 2.68559317459148088107265259495, 3.67465967626637059888483013426, 4.68085289895640541727176438744, 5.96053706999823549901230430620, 6.86995946153389996326926008449, 7.39527366704529346084038574123, 8.162727912242060737507520843938, 9.506276679685842714451361288185, 10.38052756657648434473830580809