L(s) = 1 | + (−0.707 + 0.707i)2-s + (−2.14 + 2.14i)3-s − 1.00i·4-s + (−1.71 + 1.43i)5-s − 3.04i·6-s + (0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s − 6.24i·9-s + (0.198 − 2.22i)10-s + (0.280 − 3.30i)11-s + (2.14 + 2.14i)12-s + (3.30 + 3.30i)13-s + 1.00i·14-s + (0.602 − 6.77i)15-s − 1.00·16-s + (2.29 − 2.29i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (−1.24 + 1.24i)3-s − 0.500i·4-s + (−0.766 + 0.641i)5-s − 1.24i·6-s + (0.267 − 0.267i)7-s + (0.250 + 0.250i)8-s − 2.08i·9-s + (0.0626 − 0.704i)10-s + (0.0844 − 0.996i)11-s + (0.620 + 0.620i)12-s + (0.916 + 0.916i)13-s + 0.267i·14-s + (0.155 − 1.74i)15-s − 0.250·16-s + (0.556 − 0.556i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.394 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.534807 + 0.352567i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.534807 + 0.352567i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (1.71 - 1.43i)T \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
| 11 | \( 1 + (-0.280 + 3.30i)T \) |
good | 3 | \( 1 + (2.14 - 2.14i)T - 3iT^{2} \) |
| 13 | \( 1 + (-3.30 - 3.30i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.29 + 2.29i)T - 17iT^{2} \) |
| 19 | \( 1 + 0.396T + 19T^{2} \) |
| 23 | \( 1 + (-5.93 + 5.93i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.95T + 29T^{2} \) |
| 31 | \( 1 - 3.24T + 31T^{2} \) |
| 37 | \( 1 + (5.43 + 5.43i)T + 37iT^{2} \) |
| 41 | \( 1 - 6.00iT - 41T^{2} \) |
| 43 | \( 1 + (-7.83 - 7.83i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.29 + 3.29i)T + 47iT^{2} \) |
| 53 | \( 1 + (9.91 - 9.91i)T - 53iT^{2} \) |
| 59 | \( 1 + 12.4iT - 59T^{2} \) |
| 61 | \( 1 + 7.90iT - 61T^{2} \) |
| 67 | \( 1 + (-8.03 - 8.03i)T + 67iT^{2} \) |
| 71 | \( 1 - 6.66T + 71T^{2} \) |
| 73 | \( 1 + (-3.89 - 3.89i)T + 73iT^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 + (1.89 + 1.89i)T + 83iT^{2} \) |
| 89 | \( 1 + 5.86iT - 89T^{2} \) |
| 97 | \( 1 + (-3.24 - 3.24i)T + 97iT^{2} \) |
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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
−10.74644887713609375707785263009, −9.674666256888731258287001814153, −8.910502309872944563130280048384, −7.935837207754861673063388193150, −6.71890959741784470246560734743, −6.22367395556436196449359469656, −5.12153745713889956415881945862, −4.26391292702108789425826654629, −3.32144557407222704158068452920, −0.68279495006822838197642367626,
0.893870128067532688617839367952, 1.78385870687173394251668565406, 3.56082845652816357841013099072, 4.97040121033632905131143889818, 5.67551590579653433722520322147, 6.89343173311861280978250306858, 7.63088018976214612353807945830, 8.225629803576100742505327932376, 9.259667298189681922516231893925, 10.52368842200368465385872568895