| L(s) = 1 | + (−1.95 − 6.02i)2-s + (7.28 + 5.29i)3-s + (−6.55 + 4.75i)4-s + (8.65 − 26.6i)5-s + (17.6 − 54.2i)6-s + (121. − 88.3i)7-s + (−122. − 88.9i)8-s + (25.0 + 77.0i)9-s − 177.·10-s + (−364. − 167. i)11-s − 72.8·12-s + (−95.6 − 294. i)13-s + (−769. − 559. i)14-s + (203. − 148. i)15-s + (−376. + 1.15e3i)16-s + (279. − 861. i)17-s + ⋯ |
| L(s) = 1 | + (−0.345 − 1.06i)2-s + (0.467 + 0.339i)3-s + (−0.204 + 0.148i)4-s + (0.154 − 0.476i)5-s + (0.199 − 0.614i)6-s + (0.937 − 0.681i)7-s + (−0.676 − 0.491i)8-s + (0.103 + 0.317i)9-s − 0.560·10-s + (−0.908 − 0.416i)11-s − 0.146·12-s + (−0.156 − 0.483i)13-s + (−1.04 − 0.762i)14-s + (0.233 − 0.169i)15-s + (−0.367 + 1.13i)16-s + (0.234 − 0.722i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.830336 - 1.36207i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.830336 - 1.36207i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-7.28 - 5.29i)T \) |
| 11 | \( 1 + (364. + 167. i)T \) |
| good | 2 | \( 1 + (1.95 + 6.02i)T + (-25.8 + 18.8i)T^{2} \) |
| 5 | \( 1 + (-8.65 + 26.6i)T + (-2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (-121. + 88.3i)T + (5.19e3 - 1.59e4i)T^{2} \) |
| 13 | \( 1 + (95.6 + 294. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-279. + 861. i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-1.37e3 - 999. i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 - 2.87e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (2.01e3 - 1.46e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-372. - 1.14e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (9.82e3 - 7.13e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-1.33e4 - 9.71e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 - 1.39e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.20e4 + 8.77e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-5.15e3 - 1.58e4i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (3.02e4 - 2.19e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-6.31e3 + 1.94e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 - 1.01e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-5.25e3 + 1.61e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-5.65e4 + 4.10e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (1.46e4 + 4.51e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (6.98e3 - 2.14e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 - 2.53e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (5.22e4 + 1.60e5i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−15.26524233055801259710663387951, −13.93766857304331570458141142393, −12.70046426405921623337238050997, −11.23876938553755567414196625533, −10.37053447465846281093912956002, −9.118665421084453904121170753142, −7.67970058831324168014391773288, −5.09369324742005507591090121742, −3.07751495053032257446357971454, −1.14473159623049362014203408560,
2.46808160552993138836697130136, 5.36903787370220831920472640519, 6.99472688899628530242850333582, 8.005197048385085072716399235461, 9.175368452133588154201081423367, 11.09588094644105802400276953765, 12.51173081423258675781666339728, 14.19070737346597923909847723301, 14.97636847193886654298128480053, 15.86791095754319549144966258203
