L(s) = 1 | + (−0.923 + 0.382i)3-s + (−0.705 + 1.70i)5-s + (−3.24 − 3.24i)7-s + (0.707 − 0.707i)9-s + (3.38 + 1.40i)11-s + (0.503 + 1.21i)13-s − 1.84i·15-s − 0.622i·17-s + (−2.14 − 5.17i)19-s + (4.23 + 1.75i)21-s + (2.47 − 2.47i)23-s + (1.13 + 1.13i)25-s + (−0.382 + 0.923i)27-s + (2.16 − 0.897i)29-s + 10.4·31-s + ⋯ |
L(s) = 1 | + (−0.533 + 0.220i)3-s + (−0.315 + 0.762i)5-s + (−1.22 − 1.22i)7-s + (0.235 − 0.235i)9-s + (1.02 + 0.423i)11-s + (0.139 + 0.337i)13-s − 0.476i·15-s − 0.151i·17-s + (−0.491 − 1.18i)19-s + (0.924 + 0.382i)21-s + (0.516 − 0.516i)23-s + (0.226 + 0.226i)25-s + (−0.0736 + 0.177i)27-s + (0.402 − 0.166i)29-s + 1.87·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.00011 - 0.216442i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.00011 - 0.216442i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.923 - 0.382i)T \) |
good | 5 | \( 1 + (0.705 - 1.70i)T + (-3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (3.24 + 3.24i)T + 7iT^{2} \) |
| 11 | \( 1 + (-3.38 - 1.40i)T + (7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (-0.503 - 1.21i)T + (-9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + 0.622iT - 17T^{2} \) |
| 19 | \( 1 + (2.14 + 5.17i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-2.47 + 2.47i)T - 23iT^{2} \) |
| 29 | \( 1 + (-2.16 + 0.897i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + (-0.0714 + 0.172i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-8.50 + 8.50i)T - 41iT^{2} \) |
| 43 | \( 1 + (3.62 + 1.50i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + 5.02iT - 47T^{2} \) |
| 53 | \( 1 + (-7.15 - 2.96i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (1.52 - 3.68i)T + (-41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (-3.07 + 1.27i)T + (43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (2.17 - 0.901i)T + (47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (-1.11 - 1.11i)T + 71iT^{2} \) |
| 73 | \( 1 + (-3.71 + 3.71i)T - 73iT^{2} \) |
| 79 | \( 1 - 10.2iT - 79T^{2} \) |
| 83 | \( 1 + (4.69 + 11.3i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (3.54 + 3.54i)T + 89iT^{2} \) |
| 97 | \( 1 - 0.139T + 97T^{2} \) |
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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
−10.34575262916873781226933845657, −9.622026033569134209219482944131, −8.731204569647475582144415575740, −7.08696008318947677279500902809, −6.95652747549358562019581423947, −6.16580249279447627978945253848, −4.56478081779527637277847900584, −3.89996650911634404847191480079, −2.82079795957737191775936343242, −0.71257592426231172653456191445,
1.08247139681727881189071599096, 2.79146329709640162380715486364, 3.95715018521571938973756609294, 5.12940063354709992844160983032, 6.19165521779763314072467414700, 6.44792846825539842408700824560, 7.993422642410175926710836174158, 8.701647608245542552478691453582, 9.468933559819003580416822611462, 10.27250080415992660827714723232