| L(s) = 1 | + (2.32 − 7.15i)2-s + (4.56 + 14.9i)3-s + (−19.8 − 14.4i)4-s + (28.0 − 9.12i)5-s + (117. + 1.98i)6-s + (146. − 201. i)7-s + (45.0 − 32.7i)8-s + (−201. + 136. i)9-s − 222. i·10-s + (−183. + 356. i)11-s + (124. − 362. i)12-s + (277. + 90.1i)13-s + (−1.10e3 − 1.51e3i)14-s + (264. + 377. i)15-s + (−372. − 1.14e3i)16-s + (322. + 993. i)17-s + ⋯ |
| L(s) = 1 | + (0.410 − 1.26i)2-s + (0.292 + 0.956i)3-s + (−0.621 − 0.451i)4-s + (0.502 − 0.163i)5-s + (1.32 + 0.0225i)6-s + (1.12 − 1.55i)7-s + (0.248 − 0.180i)8-s + (−0.828 + 0.560i)9-s − 0.702i·10-s + (−0.457 + 0.889i)11-s + (0.249 − 0.726i)12-s + (0.455 + 0.147i)13-s + (−1.50 − 2.06i)14-s + (0.303 + 0.432i)15-s + (−0.363 − 1.12i)16-s + (0.270 + 0.833i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.98966 - 1.27855i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.98966 - 1.27855i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.56 - 14.9i)T \) |
| 11 | \( 1 + (183. - 356. i)T \) |
| good | 2 | \( 1 + (-2.32 + 7.15i)T + (-25.8 - 18.8i)T^{2} \) |
| 5 | \( 1 + (-28.0 + 9.12i)T + (2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (-146. + 201. i)T + (-5.19e3 - 1.59e4i)T^{2} \) |
| 13 | \( 1 + (-277. - 90.1i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-322. - 993. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (986. + 1.35e3i)T + (-7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 - 3.49e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (2.33e3 + 1.69e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (577. - 1.77e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-2.27e3 - 1.65e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (902. - 656. i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 9.31e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-293. - 403. i)T + (-7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (6.54e3 + 2.12e3i)T + (3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (7.96e3 - 1.09e4i)T + (-2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (2.54e4 - 8.27e3i)T + (6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 - 3.78e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-4.70e4 + 1.52e4i)T + (1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-4.00e4 + 5.51e4i)T + (-6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (3.08e4 + 1.00e4i)T + (2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (3.04e3 + 9.37e3i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 - 7.81e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-8.74e3 + 2.69e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−15.20824738887787199237440358212, −13.93072839486428971110387527657, −13.17778502508901977531999501774, −11.33389994282753136080184151510, −10.62090150169707736312920038874, −9.651174292920499463353121869404, −7.71436367811574174771562517198, −4.85855844383917098734433248802, −3.82205884249795632311001463653, −1.72938190344366220928821910464,
2.19379515443747329330955314161, 5.43256037419770073216205300710, 6.23124325013626534596296653135, 7.952702466224508904615314371691, 8.671779332639975608889349379321, 11.19776245424561619257199161774, 12.58508819037827557870283512506, 13.96943334001804476307155400640, 14.55586640628660447578260628055, 15.66913377160972549361495927941
