L(s) = 1 | + (−1.41 + 1.41i)5-s − 3i·9-s + (4.24 + 4.24i)13-s − 2·17-s + 0.999i·25-s + (7.07 + 7.07i)29-s + (−1.41 + 1.41i)37-s + 10i·41-s + (4.24 + 4.24i)45-s + 7·49-s + (−9.89 + 9.89i)53-s + (7.07 + 7.07i)61-s − 12·65-s − 6i·73-s − 9·81-s + ⋯ |
L(s) = 1 | + (−0.632 + 0.632i)5-s − i·9-s + (1.17 + 1.17i)13-s − 0.485·17-s + 0.199i·25-s + (1.31 + 1.31i)29-s + (−0.232 + 0.232i)37-s + 1.56i·41-s + (0.632 + 0.632i)45-s + 49-s + (−1.35 + 1.35i)53-s + (0.905 + 0.905i)61-s − 1.48·65-s − 0.702i·73-s − 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.311028777\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.311028777\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + 3iT^{2} \) |
| 5 | \( 1 + (1.41 - 1.41i)T - 5iT^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 11iT^{2} \) |
| 13 | \( 1 + (-4.24 - 4.24i)T + 13iT^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 19iT^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + (-7.07 - 7.07i)T + 29iT^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (1.41 - 1.41i)T - 37iT^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (9.89 - 9.89i)T - 53iT^{2} \) |
| 59 | \( 1 - 59iT^{2} \) |
| 61 | \( 1 + (-7.07 - 7.07i)T + 61iT^{2} \) |
| 67 | \( 1 + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 + 10iT - 89T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
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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
−10.16139128814149425587196168100, −9.076347707086663046867651747910, −8.646523639204235725469867891724, −7.47012738793946725414298448013, −6.64852332190938065200837239655, −6.16233577150166182901369625795, −4.67193218998301228031221226644, −3.78608733870408137973752032025, −2.98409860813838330230526191388, −1.32276505358666812309793191908,
0.67297714011208550228282750060, 2.26810403127706478305609703346, 3.58169573353554352089447053638, 4.52836039929552121724388898178, 5.38397973208912803955262236637, 6.32348565710107898693189810229, 7.49466117160240597812517696293, 8.320156242371300399377007868027, 8.578117993620552198930777780011, 9.901994054558581508874724114738