| L(s) = 1 | + (−1.95 − 2.69i)2-s + (−1.60 − 4.94i)3-s + (1.52 − 4.68i)4-s + (6.63 + 4.81i)5-s + (−10.1 + 13.9i)6-s + (−71.1 − 23.1i)7-s + (−66.2 + 21.5i)8-s + (−21.8 + 15.8i)9-s − 27.2i·10-s + (118. + 25.9i)11-s − 25.6·12-s + (−135. − 186. i)13-s + (76.9 + 236. i)14-s + (13.1 − 40.5i)15-s + (123. + 89.8i)16-s + (106. − 146. i)17-s + ⋯ |
| L(s) = 1 | + (−0.488 − 0.673i)2-s + (−0.178 − 0.549i)3-s + (0.0951 − 0.292i)4-s + (0.265 + 0.192i)5-s + (−0.282 + 0.388i)6-s + (−1.45 − 0.472i)7-s + (−1.03 + 0.336i)8-s + (−0.269 + 0.195i)9-s − 0.272i·10-s + (0.976 + 0.214i)11-s − 0.177·12-s + (−0.801 − 1.10i)13-s + (0.392 + 1.20i)14-s + (0.0585 − 0.180i)15-s + (0.483 + 0.351i)16-s + (0.368 − 0.506i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.167i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0635732 - 0.755998i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0635732 - 0.755998i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.60 + 4.94i)T \) |
| 11 | \( 1 + (-118. - 25.9i)T \) |
| good | 2 | \( 1 + (1.95 + 2.69i)T + (-4.94 + 15.2i)T^{2} \) |
| 5 | \( 1 + (-6.63 - 4.81i)T + (193. + 594. i)T^{2} \) |
| 7 | \( 1 + (71.1 + 23.1i)T + (1.94e3 + 1.41e3i)T^{2} \) |
| 13 | \( 1 + (135. + 186. i)T + (-8.82e3 + 2.71e4i)T^{2} \) |
| 17 | \( 1 + (-106. + 146. i)T + (-2.58e4 - 7.94e4i)T^{2} \) |
| 19 | \( 1 + (-436. + 141. i)T + (1.05e5 - 7.66e4i)T^{2} \) |
| 23 | \( 1 - 331.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (1.15e3 + 375. i)T + (5.72e5 + 4.15e5i)T^{2} \) |
| 31 | \( 1 + (-675. + 490. i)T + (2.85e5 - 8.78e5i)T^{2} \) |
| 37 | \( 1 + (378. - 1.16e3i)T + (-1.51e6 - 1.10e6i)T^{2} \) |
| 41 | \( 1 + (-2.03e3 + 662. i)T + (2.28e6 - 1.66e6i)T^{2} \) |
| 43 | \( 1 + 630. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (-372. - 1.14e3i)T + (-3.94e6 + 2.86e6i)T^{2} \) |
| 53 | \( 1 + (-470. + 341. i)T + (2.43e6 - 7.50e6i)T^{2} \) |
| 59 | \( 1 + (-148. + 458. i)T + (-9.80e6 - 7.12e6i)T^{2} \) |
| 61 | \( 1 + (3.02e3 - 4.16e3i)T + (-4.27e6 - 1.31e7i)T^{2} \) |
| 67 | \( 1 - 3.90e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (-391. - 284. i)T + (7.85e6 + 2.41e7i)T^{2} \) |
| 73 | \( 1 + (4.47e3 + 1.45e3i)T + (2.29e7 + 1.66e7i)T^{2} \) |
| 79 | \( 1 + (449. + 618. i)T + (-1.20e7 + 3.70e7i)T^{2} \) |
| 83 | \( 1 + (3.89e3 - 5.36e3i)T + (-1.46e7 - 4.51e7i)T^{2} \) |
| 89 | \( 1 + 1.24e4T + 6.27e7T^{2} \) |
| 97 | \( 1 + (-1.17e4 + 8.53e3i)T + (2.73e7 - 8.41e7i)T^{2} \) |
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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
−15.41231121671457935411532559732, −14.01413450611854163997640985323, −12.71583454024168492900539565497, −11.62006869535340210912503792159, −10.12501701932583893049994473065, −9.423860433625767989540136610794, −7.20064944546125155266582384646, −5.89299589812705026870193076122, −2.90103498331536407313049266182, −0.64920208463748389258230889326,
3.45928707731563507794541270256, 5.89532682801646410066646177608, 7.11576016984359396511878313407, 9.139460697295686937259502185063, 9.562107629536172285618724647832, 11.70333102194713110105476639463, 12.72945861197886000074330589805, 14.46211092746531288694165273247, 15.74339445672546715544101299769, 16.59099850061866683624221024150
