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
−16.22404429071544712909514876432, −14.56311156456694512297019642151, −12.90812083959994738705199304408, −11.31275486687888516691708471677, −10.54460722348244938853282951210, −9.107652350841842808747925063738, −8.009757095876592738841397452372, −6.94589954236854696863800976738, −2.96719528818607989683227286961, −0.980455499642955404463952591318,
1.89194112539249218811957097435, 5.65045011255502569105902778210, 7.56166153748444654255019869878, 8.522838048846509313492483869218, 9.609296011928322653374826230000, 10.64736676699809748328952440643, 12.23230303329792233417260622659, 14.75508734472362895765630702068, 15.18204025339773215605903061137, 16.55988265866186667908088603761