L(s) = 1 | + (−0.651 − 1.12i)2-s + (0.151 − 0.262i)4-s + (−0.5 + 0.866i)5-s + (−2.30 − 3.98i)7-s − 3·8-s + 1.30·10-s + (1.30 + 2.25i)11-s + (0.302 − 0.524i)13-s + (−2.99 + 5.19i)14-s + (1.65 + 2.86i)16-s − 5.60·17-s − 3.60·19-s + (0.151 + 0.262i)20-s + (1.69 − 2.93i)22-s + (1.5 − 2.59i)23-s + ⋯ |
L(s) = 1 | + (−0.460 − 0.797i)2-s + (0.0756 − 0.131i)4-s + (−0.223 + 0.387i)5-s + (−0.870 − 1.50i)7-s − 1.06·8-s + 0.411·10-s + (0.392 + 0.680i)11-s + (0.0839 − 0.145i)13-s + (−0.801 + 1.38i)14-s + (0.412 + 0.715i)16-s − 1.35·17-s − 0.827·19-s + (0.0338 + 0.0586i)20-s + (0.361 − 0.626i)22-s + (0.312 − 0.541i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0860045 + 0.487755i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0860045 + 0.487755i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.651 + 1.12i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (2.30 + 3.98i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.30 - 2.25i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.302 + 0.524i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 5.60T + 17T^{2} \) |
| 19 | \( 1 + 3.60T + 19T^{2} \) |
| 23 | \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.30 + 7.45i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.802 - 1.39i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (1.30 - 2.25i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.30 - 5.72i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.60 + 4.51i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 5.60T + 53T^{2} \) |
| 59 | \( 1 + (-4.30 + 7.45i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.10 + 8.84i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.60 + 13.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 - 5.39T + 73T^{2} \) |
| 79 | \( 1 + (-2.19 - 3.80i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.5 + 2.59i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 7.81T + 89T^{2} \) |
| 97 | \( 1 + (4 + 6.92i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.90668471562416583825969245967, −9.886923109578972051331427497664, −9.400069120674695280008449132996, −8.043530686282579189747071781600, −6.76828947489300356315658884856, −6.41195756929778225478274790703, −4.46530699105546447482071695156, −3.49082286345633670102895803490, −2.11434378698500504246132114157, −0.34745781495859276340737750203,
2.46931585992912679144476273361, 3.70194114765628285963908975417, 5.40313969744825055379971540953, 6.24351754004800812373306090991, 6.98408993641879599244853822862, 8.295758433577730540670267931397, 8.993998823738601694438611614095, 9.288975613859666213017275099370, 10.97796316188475717661551525648, 11.81281299072388504663240378080