| L(s) = 1 | + (1.22 + 0.709i)2-s + (−2.63 + 1.32i)3-s + (0.992 + 1.73i)4-s + (−1.14 − 2.57i)5-s + (−4.15 − 0.247i)6-s + (−1.33 − 0.799i)7-s + (−0.0177 + 2.82i)8-s + (3.38 − 4.55i)9-s + (0.430 − 3.95i)10-s + (−2.39 − 1.87i)11-s + (−4.91 − 3.25i)12-s + (4.26 − 4.05i)13-s + (−1.06 − 1.92i)14-s + (6.40 + 5.25i)15-s + (−2.02 + 3.44i)16-s + (1.84 + 2.24i)17-s + ⋯ |
| L(s) = 1 | + (0.864 + 0.501i)2-s + (−1.51 + 0.764i)3-s + (0.496 + 0.868i)4-s + (−0.510 − 1.15i)5-s + (−1.69 − 0.100i)6-s + (−0.504 − 0.302i)7-s + (−0.00628 + 0.999i)8-s + (1.12 − 1.51i)9-s + (0.136 − 1.25i)10-s + (−0.722 − 0.563i)11-s + (−1.41 − 0.939i)12-s + (1.18 − 1.12i)13-s + (−0.284 − 0.514i)14-s + (1.65 + 1.35i)15-s + (−0.507 + 0.861i)16-s + (0.447 + 0.545i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 + 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.799167 - 0.345699i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.799167 - 0.345699i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.22 - 0.709i)T \) |
| good | 3 | \( 1 + (2.63 - 1.32i)T + (1.78 - 2.40i)T^{2} \) |
| 5 | \( 1 + (1.14 + 2.57i)T + (-3.35 + 3.70i)T^{2} \) |
| 7 | \( 1 + (1.33 + 0.799i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (2.39 + 1.87i)T + (2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (-4.26 + 4.05i)T + (0.637 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-1.84 - 2.24i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (0.669 + 3.86i)T + (-17.8 + 6.40i)T^{2} \) |
| 23 | \( 1 + (5.11 + 2.41i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (-4.36 - 2.47i)T + (14.9 + 24.8i)T^{2} \) |
| 31 | \( 1 + (-4.08 + 0.812i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (8.89 + 5.63i)T + (15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (3.02 + 3.34i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (3.70 + 11.2i)T + (-34.5 + 25.6i)T^{2} \) |
| 47 | \( 1 + (-9.85 - 5.26i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (6.42 - 3.64i)T + (27.2 - 45.4i)T^{2} \) |
| 59 | \( 1 + (-0.833 + 0.875i)T + (-2.89 - 58.9i)T^{2} \) |
| 61 | \( 1 + (0.227 + 3.08i)T + (-60.3 + 8.95i)T^{2} \) |
| 67 | \( 1 + (1.83 + 1.58i)T + (9.83 + 66.2i)T^{2} \) |
| 71 | \( 1 + (-4.39 - 0.652i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (6.08 - 3.64i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (-1.97 - 0.599i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (-2.25 + 1.42i)T + (35.4 - 75.0i)T^{2} \) |
| 89 | \( 1 + (-1.31 + 0.622i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (14.0 + 9.41i)T + (37.1 + 89.6i)T^{2} \) |
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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
−10.71191930414170781231879128770, −10.45889243935664858837024936614, −8.780241218278507845131727302759, −8.072494086689156645869970533696, −6.70329236989262088772036022621, −5.77221161217013920124315561278, −5.25710049219104267816842872669, −4.30118298291235475851012740288, −3.45655346110019098719632046018, −0.47656640708168703510344251262,
1.64102553967966934282506150506, 3.11090440734933322447501706303, 4.37747473273579874921166231600, 5.55828062121736998959910650056, 6.43794331507251652546794322354, 6.77849169201893983433905099968, 7.894625108469547730945313061622, 9.866268456379179545098670671150, 10.49839788273241923115041975395, 11.29678245856843226932259768183
