| L(s) = 1 | + (−7.29 + 2.37i)2-s + (4.20 + 3.05i)3-s + (34.6 − 25.1i)4-s + (2.18 − 6.71i)5-s + (−37.9 − 12.3i)6-s + (28.6 + 39.4i)7-s + (−121. + 166. i)8-s + (8.34 + 25.6i)9-s + 54.1i·10-s + (−31.5 + 116. i)11-s + 222.·12-s + (5.63 − 1.83i)13-s + (−302. − 219. i)14-s + (29.6 − 21.5i)15-s + (276. − 850. i)16-s + (404. + 131. i)17-s + ⋯ |
| L(s) = 1 | + (−1.82 + 0.592i)2-s + (0.467 + 0.339i)3-s + (2.16 − 1.57i)4-s + (0.0872 − 0.268i)5-s + (−1.05 − 0.342i)6-s + (0.585 + 0.805i)7-s + (−1.89 + 2.60i)8-s + (0.103 + 0.317i)9-s + 0.541i·10-s + (−0.260 + 0.965i)11-s + 1.54·12-s + (0.0333 − 0.0108i)13-s + (−1.54 − 1.12i)14-s + (0.131 − 0.0958i)15-s + (1.07 − 3.32i)16-s + (1.39 + 0.454i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0720 - 0.997i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0720 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.506239 + 0.544116i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.506239 + 0.544116i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.20 - 3.05i)T \) |
| 11 | \( 1 + (31.5 - 116. i)T \) |
| good | 2 | \( 1 + (7.29 - 2.37i)T + (12.9 - 9.40i)T^{2} \) |
| 5 | \( 1 + (-2.18 + 6.71i)T + (-505. - 367. i)T^{2} \) |
| 7 | \( 1 + (-28.6 - 39.4i)T + (-741. + 2.28e3i)T^{2} \) |
| 13 | \( 1 + (-5.63 + 1.83i)T + (2.31e4 - 1.67e4i)T^{2} \) |
| 17 | \( 1 + (-404. - 131. i)T + (6.75e4 + 4.90e4i)T^{2} \) |
| 19 | \( 1 + (347. - 477. i)T + (-4.02e4 - 1.23e5i)T^{2} \) |
| 23 | \( 1 - 361.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (755. + 1.03e3i)T + (-2.18e5 + 6.72e5i)T^{2} \) |
| 31 | \( 1 + (36.4 + 112. i)T + (-7.47e5 + 5.42e5i)T^{2} \) |
| 37 | \( 1 + (141. - 102. i)T + (5.79e5 - 1.78e6i)T^{2} \) |
| 41 | \( 1 + (-1.11e3 + 1.53e3i)T + (-8.73e5 - 2.68e6i)T^{2} \) |
| 43 | \( 1 + 1.41e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (177. + 128. i)T + (1.50e6 + 4.64e6i)T^{2} \) |
| 53 | \( 1 + (-349. - 1.07e3i)T + (-6.38e6 + 4.63e6i)T^{2} \) |
| 59 | \( 1 + (193. - 140. i)T + (3.74e6 - 1.15e7i)T^{2} \) |
| 61 | \( 1 + (557. + 181. i)T + (1.12e7 + 8.13e6i)T^{2} \) |
| 67 | \( 1 - 1.24e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (-472. + 1.45e3i)T + (-2.05e7 - 1.49e7i)T^{2} \) |
| 73 | \( 1 + (4.92e3 + 6.77e3i)T + (-8.77e6 + 2.70e7i)T^{2} \) |
| 79 | \( 1 + (659. - 214. i)T + (3.15e7 - 2.28e7i)T^{2} \) |
| 83 | \( 1 + (-5.68e3 - 1.84e3i)T + (3.83e7 + 2.78e7i)T^{2} \) |
| 89 | \( 1 + 813.T + 6.27e7T^{2} \) |
| 97 | \( 1 + (206. + 634. i)T + (-7.16e7 + 5.20e7i)T^{2} \) |
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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
−16.55988265866186667908088603761, −15.18204025339773215605903061137, −14.75508734472362895765630702068, −12.23230303329792233417260622659, −10.64736676699809748328952440643, −9.609296011928322653374826230000, −8.522838048846509313492483869218, −7.56166153748444654255019869878, −5.65045011255502569105902778210, −1.89194112539249218811957097435,
0.980455499642955404463952591318, 2.96719528818607989683227286961, 6.94589954236854696863800976738, 8.009757095876592738841397452372, 9.107652350841842808747925063738, 10.54460722348244938853282951210, 11.31275486687888516691708471677, 12.90812083959994738705199304408, 14.56311156456694512297019642151, 16.22404429071544712909514876432
