L(s) = 1 | + (−2.70 − 1.12i)3-s + (−0.151 − 0.366i)5-s + (3.06 − 3.06i)7-s + (3.95 + 3.95i)9-s + (3.66 − 1.51i)11-s + (−0.780 + 1.88i)13-s + 1.16i·15-s + 4.54i·17-s + (−0.221 + 0.534i)19-s + (−11.7 + 4.86i)21-s + (4.41 + 4.41i)23-s + (3.42 − 3.42i)25-s + (−2.91 − 7.03i)27-s + (4.74 + 1.96i)29-s + 0.0539·31-s + ⋯ |
L(s) = 1 | + (−1.56 − 0.647i)3-s + (−0.0678 − 0.163i)5-s + (1.15 − 1.15i)7-s + (1.31 + 1.31i)9-s + (1.10 − 0.457i)11-s + (−0.216 + 0.522i)13-s + 0.299i·15-s + 1.10i·17-s + (−0.0508 + 0.122i)19-s + (−2.56 + 1.06i)21-s + (0.921 + 0.921i)23-s + (0.684 − 0.684i)25-s + (−0.560 − 1.35i)27-s + (0.881 + 0.365i)29-s + 0.00969·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.442 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.442 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.158867280\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.158867280\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.70 + 1.12i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (0.151 + 0.366i)T + (-3.53 + 3.53i)T^{2} \) |
| 7 | \( 1 + (-3.06 + 3.06i)T - 7iT^{2} \) |
| 11 | \( 1 + (-3.66 + 1.51i)T + (7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (0.780 - 1.88i)T + (-9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 - 4.54iT - 17T^{2} \) |
| 19 | \( 1 + (0.221 - 0.534i)T + (-13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-4.41 - 4.41i)T + 23iT^{2} \) |
| 29 | \( 1 + (-4.74 - 1.96i)T + (20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 - 0.0539T + 31T^{2} \) |
| 37 | \( 1 + (-0.330 - 0.798i)T + (-26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (-0.621 - 0.621i)T + 41iT^{2} \) |
| 43 | \( 1 + (2.06 - 0.857i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + 9.44iT - 47T^{2} \) |
| 53 | \( 1 + (-10.0 + 4.16i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (2.97 + 7.17i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-9.72 - 4.02i)T + (43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (7.53 + 3.12i)T + (47.3 + 47.3i)T^{2} \) |
| 71 | \( 1 + (2.99 - 2.99i)T - 71iT^{2} \) |
| 73 | \( 1 + (2.91 + 2.91i)T + 73iT^{2} \) |
| 79 | \( 1 - 5.74iT - 79T^{2} \) |
| 83 | \( 1 + (-1.36 + 3.30i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-2.38 + 2.38i)T - 89iT^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.23768237158434173461423017363, −8.832298298579239164999735189217, −7.969027933945117041650993321407, −6.96307532808063482072678098298, −6.59575053168018883508123549187, −5.47964510224808750039244007045, −4.68166169700868026023608937583, −3.86717899732017880094937604749, −1.61667918743200463466502080484, −0.915678475219895319754870054895,
1.07584810737702601775278411457, 2.71769956209730124581231238909, 4.37125013529237407984853851331, 4.93272932775324787927052892083, 5.62072018880143126338034368898, 6.53384436056470791120141830973, 7.36204991166992527869378264584, 8.681482108284865224576575308858, 9.314586269808644412529818897315, 10.28443428244983076408910628846