| L(s) = 1 | + (2.39 − 1.53i)2-s + (0.0464 − 0.101i)4-s + (1.22 − 0.359i)5-s + (−1.84 − 2.13i)7-s + (3.19 + 22.2i)8-s + (2.37 − 2.74i)10-s + (46.9 + 30.1i)11-s + (41.6 − 48.0i)13-s + (−7.71 − 2.26i)14-s + (42.4 + 49.0i)16-s + (43.6 + 95.5i)17-s + (21.0 − 46.0i)19-s + (0.0203 − 0.141i)20-s + 158.·22-s + (41.4 − 102. i)23-s + ⋯ |
| L(s) = 1 | + (0.847 − 0.544i)2-s + (0.00581 − 0.0127i)4-s + (0.109 − 0.0321i)5-s + (−0.0997 − 0.115i)7-s + (0.141 + 0.982i)8-s + (0.0751 − 0.0867i)10-s + (1.28 + 0.826i)11-s + (0.888 − 1.02i)13-s + (−0.147 − 0.0432i)14-s + (0.663 + 0.766i)16-s + (0.622 + 1.36i)17-s + (0.254 − 0.556i)19-s + (0.000226 − 0.00157i)20-s + 1.53·22-s + (0.375 − 0.926i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00253i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.88509 + 0.00365759i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.88509 + 0.00365759i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + (-41.4 + 102. i)T \) |
| good | 2 | \( 1 + (-2.39 + 1.53i)T + (3.32 - 7.27i)T^{2} \) |
| 5 | \( 1 + (-1.22 + 0.359i)T + (105. - 67.5i)T^{2} \) |
| 7 | \( 1 + (1.84 + 2.13i)T + (-48.8 + 339. i)T^{2} \) |
| 11 | \( 1 + (-46.9 - 30.1i)T + (552. + 1.21e3i)T^{2} \) |
| 13 | \( 1 + (-41.6 + 48.0i)T + (-312. - 2.17e3i)T^{2} \) |
| 17 | \( 1 + (-43.6 - 95.5i)T + (-3.21e3 + 3.71e3i)T^{2} \) |
| 19 | \( 1 + (-21.0 + 46.0i)T + (-4.49e3 - 5.18e3i)T^{2} \) |
| 29 | \( 1 + (-25.6 - 56.0i)T + (-1.59e4 + 1.84e4i)T^{2} \) |
| 31 | \( 1 + (-25.6 - 178. i)T + (-2.85e4 + 8.39e3i)T^{2} \) |
| 37 | \( 1 + (6.66 + 1.95i)T + (4.26e4 + 2.73e4i)T^{2} \) |
| 41 | \( 1 + (-182. + 53.5i)T + (5.79e4 - 3.72e4i)T^{2} \) |
| 43 | \( 1 + (-23.4 + 163. i)T + (-7.62e4 - 2.23e4i)T^{2} \) |
| 47 | \( 1 + 347.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (175. + 202. i)T + (-2.11e4 + 1.47e5i)T^{2} \) |
| 59 | \( 1 + (260. - 300. i)T + (-2.92e4 - 2.03e5i)T^{2} \) |
| 61 | \( 1 + (53.6 + 373. i)T + (-2.17e5 + 6.39e4i)T^{2} \) |
| 67 | \( 1 + (620. - 398. i)T + (1.24e5 - 2.73e5i)T^{2} \) |
| 71 | \( 1 + (-535. + 344. i)T + (1.48e5 - 3.25e5i)T^{2} \) |
| 73 | \( 1 + (-87.3 + 191. i)T + (-2.54e5 - 2.93e5i)T^{2} \) |
| 79 | \( 1 + (99.3 - 114. i)T + (-7.01e4 - 4.88e5i)T^{2} \) |
| 83 | \( 1 + (231. + 67.9i)T + (4.81e5 + 3.09e5i)T^{2} \) |
| 89 | \( 1 + (225. - 1.57e3i)T + (-6.76e5 - 1.98e5i)T^{2} \) |
| 97 | \( 1 + (-534. + 156. i)T + (7.67e5 - 4.93e5i)T^{2} \) |
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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
−12.22090775453793433381753767858, −11.14655685525449099233482349212, −10.26675835877762605972509789192, −8.955003722229528609919054087875, −8.000113532829153454713536726125, −6.59096428101466163115301169384, −5.40093355766330631013020562950, −4.14496702506622816970684379330, −3.25564710916906479192921499348, −1.55754779545971165569735232847,
1.16751668194497125853039462051, 3.42998305130489121543127519568, 4.43280669272629394429512227476, 5.84043527872041610186215033888, 6.39789617867931018305281688396, 7.63127533160214137588476238988, 9.154173939453310391809733440119, 9.729371007694363371337477849658, 11.31822347357445687667835729243, 11.92701162184733533590378226958
