L(s) = 1 | + (0.538 − 2.01i)2-s + (−0.965 + 0.258i)3-s + (−2.02 − 1.16i)4-s + 2.08i·6-s + (1.74 − 1.99i)7-s + (−0.495 + 0.495i)8-s + (0.866 − 0.499i)9-s + (−0.971 + 1.68i)11-s + (2.25 + 0.604i)12-s + (−1.84 − 1.84i)13-s + (−3.06 − 4.57i)14-s + (−1.60 − 2.78i)16-s + (−1.62 − 6.06i)17-s + (−0.538 − 2.01i)18-s + (−1.50 − 2.61i)19-s + ⋯ |
L(s) = 1 | + (0.381 − 1.42i)2-s + (−0.557 + 0.149i)3-s + (−1.01 − 0.584i)4-s + 0.850i·6-s + (0.658 − 0.752i)7-s + (−0.175 + 0.175i)8-s + (0.288 − 0.166i)9-s + (−0.292 + 0.507i)11-s + (0.651 + 0.174i)12-s + (−0.511 − 0.511i)13-s + (−0.819 − 1.22i)14-s + (−0.401 − 0.695i)16-s + (−0.394 − 1.47i)17-s + (−0.127 − 0.474i)18-s + (−0.345 − 0.599i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.162i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.109243 - 1.33712i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.109243 - 1.33712i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.965 - 0.258i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.74 + 1.99i)T \) |
good | 2 | \( 1 + (-0.538 + 2.01i)T + (-1.73 - i)T^{2} \) |
| 11 | \( 1 + (0.971 - 1.68i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.84 + 1.84i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.62 + 6.06i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (1.50 + 2.61i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.35 - 1.43i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 - 5.27iT - 29T^{2} \) |
| 31 | \( 1 + (5.10 + 2.94i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.241 - 0.901i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 10.2iT - 41T^{2} \) |
| 43 | \( 1 + (-6.54 + 6.54i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.75 + 1.00i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.301 - 1.12i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-3.06 + 5.30i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.50 + 3.75i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (14.3 - 3.85i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 + (-6.05 + 1.62i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-14.2 + 8.21i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.197 - 0.197i)T + 83iT^{2} \) |
| 89 | \( 1 + (-6.08 - 10.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.9 + 10.9i)T - 97iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.81829701353308936135394374747, −9.894854655154158637365608645456, −9.146351594783915960111422806471, −7.56740893102248067229587808311, −6.91270293404427236790998418805, −5.02637689586577049985748485957, −4.79682626692621795132503709720, −3.47690368276015955633289382609, −2.25833713421696833699676001456, −0.76114600530089017072697868538,
2.03604561575744473921216043954, 4.08126789002550533443854811615, 5.08188876925466371694483284416, 5.81663531948030085791898949965, 6.53406511059898238685576274865, 7.55491152414973880535542007580, 8.345331200637999864761769186354, 9.088973188706728022789549199958, 10.60283479453018655547236655278, 11.21817168291027566834284556921