L(s) = 1 | + i·2-s − 4-s + (−2.21 − 0.311i)5-s − 0.622i·7-s − i·8-s + (0.311 − 2.21i)10-s − 4.42·11-s + 1.37i·13-s + 0.622·14-s + 16-s + 6.42i·17-s + 1.37·19-s + (2.21 + 0.311i)20-s − 4.42i·22-s − i·23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.990 − 0.139i)5-s − 0.235i·7-s − 0.353i·8-s + (0.0983 − 0.700i)10-s − 1.33·11-s + 0.382i·13-s + 0.166·14-s + 0.250·16-s + 1.55i·17-s + 0.316·19-s + (0.495 + 0.0695i)20-s − 0.944i·22-s − 0.208i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8617222026\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8617222026\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.21 + 0.311i)T \) |
| 23 | \( 1 + iT \) |
good | 7 | \( 1 + 0.622iT - 7T^{2} \) |
| 11 | \( 1 + 4.42T + 11T^{2} \) |
| 13 | \( 1 - 1.37iT - 13T^{2} \) |
| 17 | \( 1 - 6.42iT - 17T^{2} \) |
| 19 | \( 1 - 1.37T + 19T^{2} \) |
| 29 | \( 1 + 4.23T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 11.8iT - 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 1.05iT - 43T^{2} \) |
| 47 | \( 1 + 11.6iT - 47T^{2} \) |
| 53 | \( 1 + 3.37iT - 53T^{2} \) |
| 59 | \( 1 - 9.61T + 59T^{2} \) |
| 61 | \( 1 - 8.66T + 61T^{2} \) |
| 67 | \( 1 + 11.8iT - 67T^{2} \) |
| 71 | \( 1 - 6.99T + 71T^{2} \) |
| 73 | \( 1 - 2.75iT - 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 2.19iT - 83T^{2} \) |
| 89 | \( 1 - 2.75T + 89T^{2} \) |
| 97 | \( 1 - 2.13iT - 97T^{2} \) |
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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
−8.792729440951511451157392055598, −8.237419480449550282708025167841, −7.49802198126364310353658405369, −7.03770954648844377602102288100, −5.83081864814131854321244004924, −5.24622788684724825771832256917, −4.12628321113771518758873391094, −3.65268206130417979035410139416, −2.18493682032061904331389756914, −0.43151681621279084065412340451,
0.834379491212814437707243485570, 2.56850668613867949787631378420, 3.04952090486663413690611195635, 4.15807664497077829764616166320, 5.01763847354147373317985673322, 5.68235465023114001461456820818, 7.09564752602189414063142914811, 7.66458106727542157615892950247, 8.331078078970869198136357751587, 9.241786713562984914471682542328