| L(s) = 1 | + (−4.5 − 7.79i)3-s + (−30.2 + 52.4i)5-s + (−76.1 − 104. i)7-s + (−40.5 + 70.1i)9-s + (−152. − 263. i)11-s + 600.·13-s + 545.·15-s + (−674. − 1.16e3i)17-s + (1.16e3 − 2.01e3i)19-s + (−474. + 1.06e3i)21-s + (−438. + 759. i)23-s + (−271. − 470. i)25-s + 729·27-s − 3.24e3·29-s + (1.80e3 + 3.12e3i)31-s + ⋯ |
| L(s) = 1 | + (−0.288 − 0.499i)3-s + (−0.541 + 0.938i)5-s + (−0.587 − 0.809i)7-s + (−0.166 + 0.288i)9-s + (−0.379 − 0.656i)11-s + 0.985·13-s + 0.625·15-s + (−0.566 − 0.980i)17-s + (0.738 − 1.27i)19-s + (−0.234 + 0.527i)21-s + (−0.172 + 0.299i)23-s + (−0.0869 − 0.150i)25-s + 0.192·27-s − 0.717·29-s + (0.337 + 0.584i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.166 - 0.985i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.166 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4512270243\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4512270243\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.5 + 7.79i)T \) |
| 7 | \( 1 + (76.1 + 104. i)T \) |
| good | 5 | \( 1 + (30.2 - 52.4i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (152. + 263. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 - 600.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (674. + 1.16e3i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-1.16e3 + 2.01e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (438. - 759. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + 3.24e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (-1.80e3 - 3.12e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (3.46e3 - 6.00e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + 2.86e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.48e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (3.30e3 - 5.71e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (1.25e4 + 2.16e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (1.44e4 + 2.50e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (2.19e4 - 3.80e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-2.51e4 - 4.35e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 6.47e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (-1.13e4 - 1.95e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (1.29e4 - 2.24e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 5.57e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (3.99e4 - 6.92e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 1.16e5T + 8.58e9T^{2} \) |
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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
−11.25944050988308602977723521270, −10.28105737519303481554080557427, −9.107385611629264644498754061222, −7.902727810941532061475244418373, −7.01249476802847009864278176434, −6.48821111450360842634039473473, −5.11592106563231892098796814018, −3.62361086968573899487977545842, −2.82509319599321399721576628497, −0.959110157660739937467882094022,
0.14907409110820829276635513666, 1.75522785176207155036357987050, 3.45705606909046642748095524723, 4.39581596446899964686284296570, 5.52107211422792871962264657832, 6.31828962625129776691340245876, 7.86400295884421922597321977558, 8.658725014243615267138564594033, 9.480977313080222666284662494049, 10.42618484949884854982555251439
