| L(s) = 1 | + (−3.30 − 2.40i)2-s + (2.27 − 4.66i)3-s + (2.69 + 8.29i)4-s + (−3.98 − 5.48i)5-s + (−18.7 + 9.96i)6-s + (−17.8 + 5.81i)7-s + (0.905 − 2.78i)8-s + (−16.6 − 21.2i)9-s + 27.7i·10-s + (27.8 − 23.5i)11-s + (44.8 + 6.32i)12-s + (24.8 − 34.2i)13-s + (73.1 + 23.7i)14-s + (−34.7 + 6.10i)15-s + (46.7 − 33.9i)16-s + (−16.8 + 12.2i)17-s + ⋯ |
| L(s) = 1 | + (−1.16 − 0.849i)2-s + (0.438 − 0.898i)3-s + (0.336 + 1.03i)4-s + (−0.356 − 0.490i)5-s + (−1.27 + 0.678i)6-s + (−0.965 + 0.313i)7-s + (0.0400 − 0.123i)8-s + (−0.615 − 0.788i)9-s + 0.877i·10-s + (0.763 − 0.646i)11-s + (1.07 + 0.152i)12-s + (0.530 − 0.730i)13-s + (1.39 + 0.453i)14-s + (−0.597 + 0.105i)15-s + (0.730 − 0.530i)16-s + (−0.240 + 0.174i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.177i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0543654 - 0.607059i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0543654 - 0.607059i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.27 + 4.66i)T \) |
| 11 | \( 1 + (-27.8 + 23.5i)T \) |
| good | 2 | \( 1 + (3.30 + 2.40i)T + (2.47 + 7.60i)T^{2} \) |
| 5 | \( 1 + (3.98 + 5.48i)T + (-38.6 + 118. i)T^{2} \) |
| 7 | \( 1 + (17.8 - 5.81i)T + (277. - 201. i)T^{2} \) |
| 13 | \( 1 + (-24.8 + 34.2i)T + (-678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (16.8 - 12.2i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-4.70 - 1.52i)T + (5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + 209. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-69.0 - 212. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-195. - 142. i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-51.4 - 158. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-25.5 + 78.6i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 255. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-136. - 44.2i)T + (8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (269. - 370. i)T + (-4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-264. + 85.9i)T + (1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (101. + 139. i)T + (-7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 - 110.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-518. - 714. i)T + (-1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (664. - 215. i)T + (3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-266. + 366. i)T + (-1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-1.04e3 + 757. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + 924. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (-252. - 183. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−16.03220067588734489102629871392, −14.27983353653475158510216885665, −12.73873246941680797682860536512, −12.01410821329545924760140171535, −10.50246695561931239324615440308, −8.914439758675157883465484759806, −8.370739877202948768284318436470, −6.43953914337429000369540239366, −2.99621102811834608025222335344, −0.794443533846945052915702049992,
3.78520975729983340258225853920, 6.44067439042033150641228000783, 7.72636175606296026109812878280, 9.288245557300912271554256703246, 9.835896617867591161908061616243, 11.39464231853446452625203581332, 13.56142631336164627787818709546, 15.02612920872832954032254647409, 15.75391389294581746663107915355, 16.66357100516089923923045928317
