L(s) = 1 | + (2 + 2i)3-s + 5i·9-s + 6·11-s + (−4 + 4i)17-s − 2i·19-s + (−4 + 4i)27-s + (12 + 12i)33-s − 6·41-s + (6 + 6i)43-s − 7i·49-s − 16·51-s + (4 − 4i)57-s − 6i·59-s + (−6 + 6i)67-s + (−12 − 12i)73-s + ⋯ |
L(s) = 1 | + (1.15 + 1.15i)3-s + 1.66i·9-s + 1.80·11-s + (−0.970 + 0.970i)17-s − 0.458i·19-s + (−0.769 + 0.769i)27-s + (2.08 + 2.08i)33-s − 0.937·41-s + (0.914 + 0.914i)43-s − i·49-s − 2.24·51-s + (0.529 − 0.529i)57-s − 0.781i·59-s + (−0.733 + 0.733i)67-s + (−1.40 − 1.40i)73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.86802 + 1.47838i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.86802 + 1.47838i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-2 - 2i)T + 3iT^{2} \) |
| 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + (4 - 4i)T - 17iT^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 - 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37iT^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + (-6 - 6i)T + 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 - 53iT^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + (6 - 6i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (12 + 12i)T + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (-2 - 2i)T + 83iT^{2} \) |
| 89 | \( 1 + 18iT - 89T^{2} \) |
| 97 | \( 1 + (12 - 12i)T - 97iT^{2} \) |
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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
−10.25381163614466082658545874441, −9.425779081099485756661880713844, −8.887518060307621287878891217239, −8.291143655202486851025422713201, −7.03374647119733636966248631813, −6.10100292478583802497800651010, −4.63007235508842515941928406316, −4.03584833542529357139115688163, −3.17188990038395561560283720699, −1.83324053635990443842041189973,
1.20021717935956052295229915724, 2.26395011621514306972175801605, 3.39150364954002385347720469881, 4.39883158464657501388715052253, 6.01957604659786520321769656905, 6.88652404487218344817009907318, 7.37593524357020860553003023585, 8.523897225331465716576539229051, 8.993858394445167204015912707519, 9.730641213958487487680704655374