L(s) = 1 | + (−0.831 + 0.555i)2-s + (−0.980 − 0.195i)3-s + (0.382 − 0.923i)4-s + (0.555 − 0.831i)5-s + (0.923 − 0.382i)6-s + (0.195 + 0.980i)8-s + (0.923 + 0.382i)9-s + i·10-s + (−0.555 + 0.831i)12-s + (−0.707 + 0.707i)15-s + (−0.707 − 0.707i)16-s + (0.785 + 0.785i)17-s + (−0.980 + 0.195i)18-s + (−0.324 + 0.216i)19-s + (−0.555 − 0.831i)20-s + ⋯ |
L(s) = 1 | + (−0.831 + 0.555i)2-s + (−0.980 − 0.195i)3-s + (0.382 − 0.923i)4-s + (0.555 − 0.831i)5-s + (0.923 − 0.382i)6-s + (0.195 + 0.980i)8-s + (0.923 + 0.382i)9-s + i·10-s + (−0.555 + 0.831i)12-s + (−0.707 + 0.707i)15-s + (−0.707 − 0.707i)16-s + (0.785 + 0.785i)17-s + (−0.980 + 0.195i)18-s + (−0.324 + 0.216i)19-s + (−0.555 − 0.831i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5633085570\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5633085570\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.831 - 0.555i)T \) |
| 3 | \( 1 + (0.980 + 0.195i)T \) |
| 5 | \( 1 + (-0.555 + 0.831i)T \) |
good | 7 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 11 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 13 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 17 | \( 1 + (-0.785 - 0.785i)T + iT^{2} \) |
| 19 | \( 1 + (0.324 - 0.216i)T + (0.382 - 0.923i)T^{2} \) |
| 23 | \( 1 + (-0.636 + 1.53i)T + (-0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 31 | \( 1 + 0.765iT - T^{2} \) |
| 37 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 41 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 47 | \( 1 + (-1.38 - 1.38i)T + iT^{2} \) |
| 53 | \( 1 + (0.360 + 1.81i)T + (-0.923 + 0.382i)T^{2} \) |
| 59 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 61 | \( 1 + (0.382 + 0.0761i)T + (0.923 + 0.382i)T^{2} \) |
| 67 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 71 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 73 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + (1 - i)T - iT^{2} \) |
| 83 | \( 1 + (-1.53 + 1.02i)T + (0.382 - 0.923i)T^{2} \) |
| 89 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−10.20112166021110365662864065613, −9.336223730049960122066925914546, −8.466286843253908714713983970327, −7.70872541846262633005002070318, −6.62914582419739990277336667158, −5.98759864803307841254874746924, −5.25734134235439586665093601918, −4.34460785101279142741306946355, −2.13349022541472722125959876556, −0.935864577919110723324970379796,
1.36507989036505711871277021616, 2.78847861580325377275382927568, 3.81167825204261864475170281853, 5.18799122950868329483236083859, 6.11089198829179025433546590814, 7.11482739472531021321192228959, 7.52766514209565026159078558523, 9.044713987991844106858042146294, 9.601705524758622060245368740273, 10.44962847862682044303771310985