| L(s) = 1 | + (−2.73 + 0.732i)2-s + (−3.25 + 5.64i)3-s + (6.92 − 4i)4-s + (−20.3 − 20.3i)5-s + (4.77 − 17.8i)6-s + (−81.4 − 21.8i)7-s + (−15.9 + 16i)8-s + (19.2 + 33.3i)9-s + (70.6 + 40.7i)10-s + (−32.2 − 120. i)11-s + 52.1i·12-s + (26.5 + 166. i)13-s + 238.·14-s + (181. − 48.6i)15-s + (31.9 − 55.4i)16-s + (−57.7 + 33.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.683 + 0.183i)2-s + (−0.362 + 0.627i)3-s + (0.433 − 0.250i)4-s + (−0.815 − 0.815i)5-s + (0.132 − 0.494i)6-s + (−1.66 − 0.445i)7-s + (−0.249 + 0.250i)8-s + (0.237 + 0.411i)9-s + (0.706 + 0.407i)10-s + (−0.266 − 0.993i)11-s + 0.362i·12-s + (0.156 + 0.987i)13-s + 1.21·14-s + (0.807 − 0.216i)15-s + (0.124 − 0.216i)16-s + (−0.199 + 0.115i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.430i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.00917888 - 0.0405597i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00917888 - 0.0405597i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.73 - 0.732i)T \) |
| 13 | \( 1 + (-26.5 - 166. i)T \) |
| good | 3 | \( 1 + (3.25 - 5.64i)T + (-40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (20.3 + 20.3i)T + 625iT^{2} \) |
| 7 | \( 1 + (81.4 + 21.8i)T + (2.07e3 + 1.20e3i)T^{2} \) |
| 11 | \( 1 + (32.2 + 120. i)T + (-1.26e4 + 7.32e3i)T^{2} \) |
| 17 | \( 1 + (57.7 - 33.3i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (119. - 445. i)T + (-1.12e5 - 6.51e4i)T^{2} \) |
| 23 | \( 1 + (309. + 178. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-600. + 1.04e3i)T + (-3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 + (356. + 356. i)T + 9.23e5iT^{2} \) |
| 37 | \( 1 + (-48.5 - 181. i)T + (-1.62e6 + 9.37e5i)T^{2} \) |
| 41 | \( 1 + (-67.2 + 18.0i)T + (2.44e6 - 1.41e6i)T^{2} \) |
| 43 | \( 1 + (492. - 284. i)T + (1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-1.67e3 + 1.67e3i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + 5.17e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (3.25e3 + 873. i)T + (1.04e7 + 6.05e6i)T^{2} \) |
| 61 | \( 1 + (-686. - 1.18e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (7.19e3 - 1.92e3i)T + (1.74e7 - 1.00e7i)T^{2} \) |
| 71 | \( 1 + (473. - 1.76e3i)T + (-2.20e7 - 1.27e7i)T^{2} \) |
| 73 | \( 1 + (-5.57e3 + 5.57e3i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 - 969.T + 3.89e7T^{2} \) |
| 83 | \( 1 + (-7.72e3 - 7.72e3i)T + 4.74e7iT^{2} \) |
| 89 | \( 1 + (-402. - 1.50e3i)T + (-5.43e7 + 3.13e7i)T^{2} \) |
| 97 | \( 1 + (4.26e3 - 1.59e4i)T + (-7.66e7 - 4.42e7i)T^{2} \) |
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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
−16.40484060603870191913142452133, −15.68144346621307397698317622023, −13.56989663153028434712059916989, −12.13507840127344262277020102192, −10.65068251278306216919499566882, −9.519414475266983175527887946080, −8.068876438389653443394932095934, −6.22597300921691583768041297626, −4.03593169350008820542391538157, −0.04089852157007801226210104919,
3.11350232727217642049454277640, 6.48802713757416859454198485575, 7.38932120485136473871526289295, 9.369654993954470775298717024313, 10.68185022130128294278157594327, 12.19761587888108742487311619661, 12.92174840303748401018646871506, 15.28390923037517075841600512155, 15.84058644914262183239751377905, 17.62095091433185190540300014469
