| L(s) = 1 | + (0.814 + 1.52i)3-s + i·5-s + i·7-s + (−1.67 + 2.49i)9-s − 1.02·11-s − 4.05·13-s + (−1.52 + 0.814i)15-s − 3.17i·17-s + 2.43i·19-s + (−1.52 + 0.814i)21-s − 6.89·23-s − 25-s + (−5.16 − 0.527i)27-s − 4.16i·29-s − 2.76i·31-s + ⋯ |
| L(s) = 1 | + (0.470 + 0.882i)3-s + 0.447i·5-s + 0.377i·7-s + (−0.557 + 0.830i)9-s − 0.310·11-s − 1.12·13-s + (−0.394 + 0.210i)15-s − 0.770i·17-s + 0.558i·19-s + (−0.333 + 0.177i)21-s − 1.43·23-s − 0.200·25-s + (−0.994 − 0.101i)27-s − 0.772i·29-s − 0.496i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.291 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.291 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01267571384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01267571384\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.814 - 1.52i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 + 1.02T + 11T^{2} \) |
| 13 | \( 1 + 4.05T + 13T^{2} \) |
| 17 | \( 1 + 3.17iT - 17T^{2} \) |
| 19 | \( 1 - 2.43iT - 19T^{2} \) |
| 23 | \( 1 + 6.89T + 23T^{2} \) |
| 29 | \( 1 + 4.16iT - 29T^{2} \) |
| 31 | \( 1 + 2.76iT - 31T^{2} \) |
| 37 | \( 1 - 0.949T + 37T^{2} \) |
| 41 | \( 1 + 5.27iT - 41T^{2} \) |
| 43 | \( 1 + 9.82iT - 43T^{2} \) |
| 47 | \( 1 - 5.55T + 47T^{2} \) |
| 53 | \( 1 - 2.52iT - 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 12.3iT - 67T^{2} \) |
| 71 | \( 1 - 0.483T + 71T^{2} \) |
| 73 | \( 1 + 2.55T + 73T^{2} \) |
| 79 | \( 1 - 4.87iT - 79T^{2} \) |
| 83 | \( 1 + 8.24T + 83T^{2} \) |
| 89 | \( 1 + 2.15iT - 89T^{2} \) |
| 97 | \( 1 - 1.82T + 97T^{2} \) |
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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
−9.220295125186960351016325379366, −8.440101939936154309895498573604, −7.70602561121470061641254735837, −7.11820717386327282293040675578, −5.87936094077458751205394043648, −5.39573239904093630796756861838, −4.40570240068699935692148840780, −3.74814725340512940114568083188, −2.63637831657561305925345753146, −2.19777989364037772323512549240,
0.00337713108168654629613581082, 1.31866096119868334520457821564, 2.25520065007181839453527358740, 3.15721208442758697427801585567, 4.18215773729915089852185688709, 5.02279626662380436664021344610, 5.98313712983281535976686143732, 6.68076640663458723844717738741, 7.52319953431477594538661763510, 7.973185251279028735632107191715
