L(s) = 1 | + (0.273 + 1.38i)2-s + (−1.55 + 0.758i)3-s + (−1.85 + 0.758i)4-s + (−1.47 − 1.95i)6-s + 3.56i·7-s + (−1.55 − 2.36i)8-s + (1.85 − 2.36i)9-s − 4.20·11-s + (2.30 − 2.58i)12-s − 2.70·13-s + (−4.94 + 0.973i)14-s + (2.85 − 2.80i)16-s − 0.828i·17-s + (3.78 + 1.92i)18-s − 5.07i·19-s + ⋯ |
L(s) = 1 | + (0.193 + 0.981i)2-s + (−0.899 + 0.437i)3-s + (−0.925 + 0.379i)4-s + (−0.603 − 0.797i)6-s + 1.34i·7-s + (−0.550 − 0.834i)8-s + (0.616 − 0.787i)9-s − 1.26·11-s + (0.666 − 0.745i)12-s − 0.749·13-s + (−1.32 + 0.260i)14-s + (0.712 − 0.701i)16-s − 0.200i·17-s + (0.891 + 0.453i)18-s − 1.16i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.180593 - 0.403469i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.180593 - 0.403469i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.273 - 1.38i)T \) |
| 3 | \( 1 + (1.55 - 0.758i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 3.56iT - 7T^{2} \) |
| 11 | \( 1 + 4.20T + 11T^{2} \) |
| 13 | \( 1 + 2.70T + 13T^{2} \) |
| 17 | \( 1 + 0.828iT - 17T^{2} \) |
| 19 | \( 1 + 5.07iT - 19T^{2} \) |
| 23 | \( 1 + 1.09T + 23T^{2} \) |
| 29 | \( 1 - 5.55iT - 29T^{2} \) |
| 31 | \( 1 - 6.59iT - 31T^{2} \) |
| 37 | \( 1 + 5.40T + 37T^{2} \) |
| 41 | \( 1 - 10.2iT - 41T^{2} \) |
| 43 | \( 1 - 0.531iT - 43T^{2} \) |
| 47 | \( 1 + 6.22T + 47T^{2} \) |
| 53 | \( 1 - 5.55iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 0.701T + 61T^{2} \) |
| 67 | \( 1 + 2.04iT - 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 7.70T + 73T^{2} \) |
| 79 | \( 1 - 7.12iT - 79T^{2} \) |
| 83 | \( 1 + 3.11T + 83T^{2} \) |
| 89 | \( 1 + 4.72iT - 89T^{2} \) |
| 97 | \( 1 - 8.10T + 97T^{2} \) |
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
−12.46938387999804855663942283290, −11.55240707621433248716000212324, −10.34968627140288824053066984520, −9.391760953981108049602591057357, −8.547940694550969760759196226830, −7.30766494515808055172426179323, −6.31793638249982643985431390050, −5.18846785511935601332943982090, −4.91097279564200273946532134827, −2.99347916384676939488650535405,
0.33049392558902920622752100175, 2.06213102508203619720255057633, 3.84986612335308103292857002092, 4.91035266434224874123427373673, 5.90823638964674845752129598823, 7.35055561556732689208212841747, 8.112593251511946482197077267825, 9.972762803431061538306726348524, 10.27935854852351420510005906004, 11.13633093880865339886363977309