| L(s) = 1 | + (1.71 − 0.227i)3-s + (−0.0496 − 2.23i)5-s + (−2.02 + 2.02i)7-s + (2.89 − 0.779i)9-s + (2.82 − 3.89i)11-s + (5.74 + 0.910i)13-s + (−0.593 − 3.82i)15-s + (−1.36 + 0.694i)17-s + (−3.53 − 1.14i)19-s + (−3.01 + 3.92i)21-s + (−2.52 + 0.400i)23-s + (−4.99 + 0.222i)25-s + (4.79 − 1.99i)27-s + (−0.0334 − 0.102i)29-s + (−3.13 + 9.63i)31-s + ⋯ |
| L(s) = 1 | + (0.991 − 0.131i)3-s + (−0.0222 − 0.999i)5-s + (−0.763 + 0.763i)7-s + (0.965 − 0.259i)9-s + (0.852 − 1.17i)11-s + (1.59 + 0.252i)13-s + (−0.153 − 0.988i)15-s + (−0.330 + 0.168i)17-s + (−0.810 − 0.263i)19-s + (−0.656 + 0.857i)21-s + (−0.526 + 0.0834i)23-s + (−0.999 + 0.0444i)25-s + (0.923 − 0.384i)27-s + (−0.00620 − 0.0191i)29-s + (−0.562 + 1.73i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.67217 - 0.497228i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67217 - 0.497228i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.71 + 0.227i)T \) |
| 5 | \( 1 + (0.0496 + 2.23i)T \) |
| good | 7 | \( 1 + (2.02 - 2.02i)T - 7iT^{2} \) |
| 11 | \( 1 + (-2.82 + 3.89i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-5.74 - 0.910i)T + (12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (1.36 - 0.694i)T + (9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (3.53 + 1.14i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (2.52 - 0.400i)T + (21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (0.0334 + 0.102i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (3.13 - 9.63i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.418 - 2.64i)T + (-35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (2.26 + 3.11i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (1.86 + 1.86i)T + 43iT^{2} \) |
| 47 | \( 1 + (4.48 - 8.80i)T + (-27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-9.81 - 4.99i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (0.210 - 0.153i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (4.36 + 3.17i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.74 - 3.42i)T + (-39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (14.1 - 4.61i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.320 + 2.02i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-8.17 + 2.65i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.16 - 6.21i)T + (-48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (3.84 + 2.79i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-14.9 - 7.63i)T + (57.0 + 78.4i)T^{2} \) |
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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
−11.86940316563732175546712129602, −10.67695269329044250115577494476, −9.296424844537813587463298463541, −8.771628406400580532211816494278, −8.343081392233940415514081968919, −6.65517915734473633920661597145, −5.84107971486920625158846600252, −4.16951277208067828121087715997, −3.23285419006640877930487140841, −1.49380992192838559995908722478,
2.03735735037457961391666708588, 3.58109729191853120962741000407, 4.09747152473321876037296183093, 6.28649734898229651008684514739, 6.96882580621110932870094957801, 7.939296879834180061333935452239, 9.101612004162593700464709705806, 10.00178789895850042419016600362, 10.60458783011223595056924770595, 11.76042391233688950546089484067
