L(s) = 1 | + (0.707 − 0.707i)2-s + (1.41 + i)3-s − 1.00i·4-s + (−1.90 + 1.17i)5-s + (1.70 − 0.292i)6-s + (2.66 + 2.66i)7-s + (−0.707 − 0.707i)8-s + (1.00 + 2.82i)9-s + (−0.512 + 2.17i)10-s − 1.41i·11-s + (1.00 − 1.41i)12-s + (1.02 − 1.02i)13-s + 3.76·14-s + (−3.86 − 0.237i)15-s − 1.00·16-s + (−0.707 + 0.707i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.816 + 0.577i)3-s − 0.500i·4-s + (−0.850 + 0.526i)5-s + (0.696 − 0.119i)6-s + (1.00 + 1.00i)7-s + (−0.250 − 0.250i)8-s + (0.333 + 0.942i)9-s + (−0.162 + 0.688i)10-s − 0.426i·11-s + (0.288 − 0.408i)12-s + (0.284 − 0.284i)13-s + 1.00·14-s + (−0.998 − 0.0613i)15-s − 0.250·16-s + (−0.171 + 0.171i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.21380 + 0.556041i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.21380 + 0.556041i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (-1.41 - i)T \) |
| 5 | \( 1 + (1.90 - 1.17i)T \) |
| 17 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 + (-2.66 - 2.66i)T + 7iT^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 + (-1.02 + 1.02i)T - 13iT^{2} \) |
| 19 | \( 1 - 5.57iT - 19T^{2} \) |
| 23 | \( 1 + (1.05 + 1.05i)T + 23iT^{2} \) |
| 29 | \( 1 - 1.85T + 29T^{2} \) |
| 31 | \( 1 - 9.92T + 31T^{2} \) |
| 37 | \( 1 + (6.88 + 6.88i)T + 37iT^{2} \) |
| 41 | \( 1 + 9.48iT - 41T^{2} \) |
| 43 | \( 1 + (-6.21 + 6.21i)T - 43iT^{2} \) |
| 47 | \( 1 + (6 - 6i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.35 + 2.35i)T + 53iT^{2} \) |
| 59 | \( 1 + 1.60T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 + (2.86 + 2.86i)T + 67iT^{2} \) |
| 71 | \( 1 + 7.81iT - 71T^{2} \) |
| 73 | \( 1 + (-4.70 + 4.70i)T - 73iT^{2} \) |
| 79 | \( 1 + 10.5iT - 79T^{2} \) |
| 83 | \( 1 + (-2.21 - 2.21i)T + 83iT^{2} \) |
| 89 | \( 1 - 3.29T + 89T^{2} \) |
| 97 | \( 1 + (0.514 + 0.514i)T + 97iT^{2} \) |
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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
−10.80801579227063072783143627093, −10.46523330686958679169749210472, −9.101189760999541113405724859105, −8.333975503989407454373479216396, −7.69861035306080446737521891385, −6.13323978492528776914858293254, −5.03042635623423302406555910717, −4.04423552047295919432232687663, −3.12857327870476155720043647187, −2.01801549976843757658821582109,
1.25110551646212572230703527924, 3.01743673466271467205626651559, 4.32625871199781424681408013171, 4.74227763329120390424765059676, 6.56755519609869396978367330998, 7.28305596771434026017110569468, 8.064235957564624676300091852655, 8.583021202269703665013871821450, 9.751720166116959550028364338349, 11.15888768291663873739414036383