| L(s) = 1 | + (−0.309 + 0.951i)2-s + (0.408 + 0.296i)3-s + (−0.809 − 0.587i)4-s + (−0.206 − 2.22i)5-s + (−0.408 + 0.296i)6-s − 7-s + (0.809 − 0.587i)8-s + (−0.848 − 2.61i)9-s + (2.18 + 0.491i)10-s + (0.608 − 1.87i)11-s + (−0.156 − 0.480i)12-s + (−0.779 − 2.40i)13-s + (0.309 − 0.951i)14-s + (0.576 − 0.971i)15-s + (0.309 + 0.951i)16-s + (−2.55 + 1.85i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (0.235 + 0.171i)3-s + (−0.404 − 0.293i)4-s + (−0.0924 − 0.995i)5-s + (−0.166 + 0.121i)6-s − 0.377·7-s + (0.286 − 0.207i)8-s + (−0.282 − 0.870i)9-s + (0.689 + 0.155i)10-s + (0.183 − 0.565i)11-s + (−0.0450 − 0.138i)12-s + (−0.216 − 0.665i)13-s + (0.0825 − 0.254i)14-s + (0.148 − 0.250i)15-s + (0.0772 + 0.237i)16-s + (−0.619 + 0.450i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 + 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.682 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.930413 - 0.404422i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.930413 - 0.404422i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (0.206 + 2.22i)T \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + (-0.408 - 0.296i)T + (0.927 + 2.85i)T^{2} \) |
| 11 | \( 1 + (-0.608 + 1.87i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (0.779 + 2.40i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (2.55 - 1.85i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-5.45 + 3.95i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.753 + 2.31i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-1.36 - 0.993i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (1.32 - 0.964i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.66 - 8.20i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (0.775 + 2.38i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 8.20T + 43T^{2} \) |
| 47 | \( 1 + (-0.413 - 0.300i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-2.70 - 1.96i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (3.41 + 10.5i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (0.274 - 0.843i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (10.5 - 7.69i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (5.71 + 4.15i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.244 + 0.752i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-4.34 - 3.16i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (3.62 - 2.63i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-4.29 + 13.2i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-6.64 - 4.82i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.42719045296105814845150010411, −10.14559429343169332536034648437, −9.163785607744085613667795448785, −8.752616221979218884287868403499, −7.70595475239075169203113580745, −6.50824910158644646697575962518, −5.56987089532088702169513294587, −4.46957453228570317255521520515, −3.16511770183992870226450089474, −0.75364913466258876825673635147,
2.02904665657229155899156515999, 3.05966497519910207359228993507, 4.30891925036724113388071155040, 5.76366191477294306401143218221, 7.16548242295485782374701792299, 7.69556071532479944585858130955, 9.079150862830935220133528017198, 9.833094742688487199953563883500, 10.74120426495699422878388567023, 11.50658861190393121532640987577
