L(s) = 1 | + (1.93 − 1.93i)3-s + (−1.73 − 1.73i)5-s + 1.03i·7-s − 4.46i·9-s + (0.896 + 0.896i)11-s + (3.73 − 3.73i)13-s − 6.69·15-s − 3.46·17-s + (−0.896 + 0.896i)19-s + (1.99 + 1.99i)21-s + 6.69i·23-s + 0.999i·25-s + (−2.82 − 2.82i)27-s + (−1.73 + 1.73i)29-s + 5.65·31-s + ⋯ |
L(s) = 1 | + (1.11 − 1.11i)3-s + (−0.774 − 0.774i)5-s + 0.391i·7-s − 1.48i·9-s + (0.270 + 0.270i)11-s + (1.03 − 1.03i)13-s − 1.72·15-s − 0.840·17-s + (−0.205 + 0.205i)19-s + (0.436 + 0.436i)21-s + 1.39i·23-s + 0.199i·25-s + (−0.544 − 0.544i)27-s + (−0.321 + 0.321i)29-s + 1.01·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.18431 - 1.03861i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.18431 - 1.03861i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.93 + 1.93i)T - 3iT^{2} \) |
| 5 | \( 1 + (1.73 + 1.73i)T + 5iT^{2} \) |
| 7 | \( 1 - 1.03iT - 7T^{2} \) |
| 11 | \( 1 + (-0.896 - 0.896i)T + 11iT^{2} \) |
| 13 | \( 1 + (-3.73 + 3.73i)T - 13iT^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + (0.896 - 0.896i)T - 19iT^{2} \) |
| 23 | \( 1 - 6.69iT - 23T^{2} \) |
| 29 | \( 1 + (1.73 - 1.73i)T - 29iT^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + (0.267 + 0.267i)T + 37iT^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 + (-5.79 - 5.79i)T + 43iT^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 + (4.26 + 4.26i)T + 53iT^{2} \) |
| 59 | \( 1 + (7.58 + 7.58i)T + 59iT^{2} \) |
| 61 | \( 1 + (0.267 - 0.267i)T - 61iT^{2} \) |
| 67 | \( 1 + (2.96 - 2.96i)T - 67iT^{2} \) |
| 71 | \( 1 - 6.69iT - 71T^{2} \) |
| 73 | \( 1 - 9.46iT - 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + (5.79 - 5.79i)T - 83iT^{2} \) |
| 89 | \( 1 + 9.46iT - 89T^{2} \) |
| 97 | \( 1 + 3.46T + 97T^{2} \) |
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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
−12.09701283569521907982806432177, −11.09019041543006503777682326697, −9.496368620207651446655541435953, −8.524447120042847340161034610161, −8.113263439399120570847420678667, −7.11887240151001153089050321082, −5.83974946098765546017082158662, −4.18729248464212392117361759137, −2.91557345670898583173198075752, −1.32108093557067939867450682948,
2.62391540854776154640244642046, 3.88349682334901297168133473186, 4.33430156856608423130778194820, 6.37148758956989615432578989638, 7.45697213592401833883878212511, 8.667048184195163602989145053399, 9.108497635137778373443061857432, 10.53997084625982200802463407387, 10.88519229997190598484957819090, 12.03681699556156043557324411526