| L(s) = 1 | + (−1.35 + 0.421i)2-s + (0.423 + 1.30i)3-s + (1.64 − 1.13i)4-s + (−1.11 + 1.93i)5-s + (−1.12 − 1.58i)6-s + 0.927i·7-s + (−1.74 + 2.22i)8-s + (0.904 − 0.657i)9-s + (0.687 − 3.08i)10-s + (−1.54 + 2.12i)11-s + (2.18 + 1.66i)12-s + (−2.08 + 1.51i)13-s + (−0.390 − 1.25i)14-s + (−3.00 − 0.631i)15-s + (1.41 − 3.74i)16-s + (0.893 + 0.290i)17-s + ⋯ |
| L(s) = 1 | + (−0.954 + 0.297i)2-s + (0.244 + 0.753i)3-s + (0.822 − 0.568i)4-s + (−0.498 + 0.867i)5-s + (−0.457 − 0.646i)6-s + 0.350i·7-s + (−0.616 + 0.787i)8-s + (0.301 − 0.219i)9-s + (0.217 − 0.976i)10-s + (−0.465 + 0.640i)11-s + (0.629 + 0.480i)12-s + (−0.578 + 0.420i)13-s + (−0.104 − 0.334i)14-s + (−0.774 − 0.163i)15-s + (0.353 − 0.935i)16-s + (0.216 + 0.0704i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 - 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.304919 + 0.639450i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.304919 + 0.639450i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.35 - 0.421i)T \) |
| 5 | \( 1 + (1.11 - 1.93i)T \) |
| good | 3 | \( 1 + (-0.423 - 1.30i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 - 0.927iT - 7T^{2} \) |
| 11 | \( 1 + (1.54 - 2.12i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (2.08 - 1.51i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.893 - 0.290i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (0.336 + 0.109i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (4.22 - 5.80i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (1.33 - 0.432i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.34 - 4.13i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-7.47 + 5.42i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-7.67 + 5.57i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 5.59T + 43T^{2} \) |
| 47 | \( 1 + (-7.85 + 2.55i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (3.01 + 9.26i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-2.55 - 3.52i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.14 + 2.95i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (0.325 - 1.00i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-3.56 - 10.9i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (9.25 - 12.7i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (1.73 + 5.32i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-2.17 + 6.70i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-7.12 - 5.17i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-1.52 + 0.496i)T + (78.4 - 57.0i)T^{2} \) |
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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
−12.49594845678996211109207767577, −11.54557588871597233616023902515, −10.54122109225350007072162540424, −9.837371229113111586841538234807, −9.039121845402338263183420775511, −7.68695402912300830905984833220, −7.03652382979437135120756364959, −5.62737433592356326735902985882, −3.99270820133411678949660198212, −2.42575062042769567594669609289,
0.825468786656442078104011103021, 2.53874605231569493184012895141, 4.31059301211106180168637877238, 6.11005307199443686494970934799, 7.67940993645362317937107603718, 7.82593786527764777607986389122, 8.988356176174557562959457725815, 10.11432833762481096740569830973, 11.07632711308818119635880799998, 12.23442669549020064154640961119
