| L(s) = 1 | + (3.83 − 2.78i)2-s + (−4.74 − 2.11i)3-s + (4.45 − 13.7i)4-s + (−2.11 + 2.91i)5-s + (−24.0 + 5.12i)6-s + (28.5 + 9.26i)7-s + (−9.40 − 28.9i)8-s + (18.0 + 20.0i)9-s + 17.0i·10-s + (−32.5 + 16.4i)11-s + (−50.1 + 55.7i)12-s + (−23.3 − 32.1i)13-s + (135. − 43.8i)14-s + (16.2 − 9.36i)15-s + (−23.2 − 16.8i)16-s + (−18.3 − 13.3i)17-s + ⋯ |
| L(s) = 1 | + (1.35 − 0.984i)2-s + (−0.913 − 0.406i)3-s + (0.557 − 1.71i)4-s + (−0.189 + 0.260i)5-s + (−1.63 + 0.348i)6-s + (1.53 + 0.500i)7-s + (−0.415 − 1.27i)8-s + (0.669 + 0.742i)9-s + 0.539i·10-s + (−0.892 + 0.451i)11-s + (−1.20 + 1.34i)12-s + (−0.498 − 0.685i)13-s + (2.57 − 0.837i)14-s + (0.279 − 0.161i)15-s + (−0.363 − 0.263i)16-s + (−0.261 − 0.190i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.42297 - 1.26323i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42297 - 1.26323i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.74 + 2.11i)T \) |
| 11 | \( 1 + (32.5 - 16.4i)T \) |
| good | 2 | \( 1 + (-3.83 + 2.78i)T + (2.47 - 7.60i)T^{2} \) |
| 5 | \( 1 + (2.11 - 2.91i)T + (-38.6 - 118. i)T^{2} \) |
| 7 | \( 1 + (-28.5 - 9.26i)T + (277. + 201. i)T^{2} \) |
| 13 | \( 1 + (23.3 + 32.1i)T + (-678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (18.3 + 13.3i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (74.0 - 24.0i)T + (5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 - 39.4iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-60.7 + 186. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (125. - 90.9i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-103. + 319. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (67.1 + 206. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 - 193. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (188. - 61.1i)T + (8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-28.7 - 39.5i)T + (-4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-105. - 34.2i)T + (1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (187. - 257. i)T + (-7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + 329.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-348. + 479. i)T + (-1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-286. - 93.1i)T + (3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (324. + 447. i)T + (-1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-508. - 369. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + 1.05e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (-402. + 292. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.43949595274179323018212301463, −14.57433119297176261321675498801, −13.15698261297887519821686407288, −12.24928214880763419352337231541, −11.26955277777623866907747905042, −10.52438505532765419477391735124, −7.71149412776810780760593861483, −5.60825419080027183184244281345, −4.67610552284762119763702193227, −2.10724316607311226928283293048,
4.39482794622454654494280870629, 5.11427627232129681336666182772, 6.71407001278893190092705059695, 8.162826693687156983841446578406, 10.69746972041168788529701753400, 11.82007699984866312572952990307, 13.02976843144928063063549601954, 14.35448077476059817651933452859, 15.20497800155069180214215954087, 16.39352785072353062954914137557
