L(s) = 1 | + (2.15 + 1.24i)5-s + 0.872i·7-s + 2.41·11-s + (−1.98 − 3.43i)13-s + (−0.303 − 0.174i)17-s + (1.35 − 4.14i)19-s + (−3.88 − 6.72i)23-s + (0.608 + 1.05i)25-s + (7.96 − 4.59i)29-s − 2.43i·31-s + (−1.08 + 1.88i)35-s + 6.18·37-s + (1.31 + 0.760i)41-s + (−5.24 − 3.02i)43-s + (−4.05 − 7.01i)47-s + ⋯ |
L(s) = 1 | + (0.965 + 0.557i)5-s + 0.329i·7-s + 0.726·11-s + (−0.549 − 0.951i)13-s + (−0.0734 − 0.0424i)17-s + (0.311 − 0.950i)19-s + (−0.809 − 1.40i)23-s + (0.121 + 0.210i)25-s + (1.47 − 0.853i)29-s − 0.437i·31-s + (−0.183 + 0.318i)35-s + 1.01·37-s + (0.205 + 0.118i)41-s + (−0.800 − 0.462i)43-s + (−0.590 − 1.02i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.510i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 + 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.181941828\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.181941828\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-1.35 + 4.14i)T \) |
good | 5 | \( 1 + (-2.15 - 1.24i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 - 0.872iT - 7T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 13 | \( 1 + (1.98 + 3.43i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.303 + 0.174i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (3.88 + 6.72i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.96 + 4.59i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 2.43iT - 31T^{2} \) |
| 37 | \( 1 - 6.18T + 37T^{2} \) |
| 41 | \( 1 + (-1.31 - 0.760i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.24 + 3.02i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.05 + 7.01i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.41 + 1.97i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.55 + 2.68i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.53 - 11.3i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.04 - 2.91i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.65 - 8.06i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (1.01 - 1.75i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-10.8 - 6.25i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 6.29T + 83T^{2} \) |
| 89 | \( 1 + (-14.0 + 8.11i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.90 - 15.4i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.739908136604596754732458459306, −8.120515439849206810207554451729, −7.06204219917343226172529276888, −6.42941428808788618356105650286, −5.81082218476155775698716120389, −4.93142687970226371709473769081, −4.00486611813238422593403242195, −2.67338662557657750351731834811, −2.35081847998921923238979044144, −0.75553853787900577549724318649,
1.26984805482518472884650114266, 1.91530079439238720180783772168, 3.24927526669696308152891349090, 4.22942953124543769556322877585, 4.98856537905715611643556833216, 5.87102403952370546517849455655, 6.49841484990933164516008356404, 7.34978177317100341135493362091, 8.179515319711913643405720421897, 9.081997150680578330820042349905