| L(s) = 1 | + (−5.10 + 3.70i)2-s + (2.78 + 8.55i)3-s + (2.39 − 7.36i)4-s + (54.7 + 39.7i)5-s + (−45.9 − 33.3i)6-s + (−37.8 + 116. i)7-s + (−47.2 − 145. i)8-s + (−65.5 + 47.6i)9-s − 426.·10-s + (−315. − 248. i)11-s + 69.6·12-s + (−341. + 248. i)13-s + (−238. − 734. i)14-s + (−188. + 578. i)15-s + (980. + 712. i)16-s + (−766. − 556. i)17-s + ⋯ |
| L(s) = 1 | + (−0.901 + 0.655i)2-s + (0.178 + 0.549i)3-s + (0.0747 − 0.230i)4-s + (0.978 + 0.710i)5-s + (−0.520 − 0.378i)6-s + (−0.291 + 0.898i)7-s + (−0.261 − 0.803i)8-s + (−0.269 + 0.195i)9-s − 1.34·10-s + (−0.786 − 0.618i)11-s + 0.139·12-s + (−0.561 + 0.407i)13-s + (−0.325 − 1.00i)14-s + (−0.215 + 0.664i)15-s + (0.957 + 0.695i)16-s + (−0.643 − 0.467i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0618i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0264653 + 0.854550i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0264653 + 0.854550i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.78 - 8.55i)T \) |
| 11 | \( 1 + (315. + 248. i)T \) |
| good | 2 | \( 1 + (5.10 - 3.70i)T + (9.88 - 30.4i)T^{2} \) |
| 5 | \( 1 + (-54.7 - 39.7i)T + (965. + 2.97e3i)T^{2} \) |
| 7 | \( 1 + (37.8 - 116. i)T + (-1.35e4 - 9.87e3i)T^{2} \) |
| 13 | \( 1 + (341. - 248. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (766. + 556. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-425. - 1.31e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 - 4.75e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (751. - 2.31e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (5.44e3 - 3.95e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (1.47e3 - 4.54e3i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (2.52e3 + 7.77e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 1.33e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-8.94e3 - 2.75e4i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-7.66e3 + 5.57e3i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-2.16e3 + 6.67e3i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-2.48e4 - 1.80e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + 2.44e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-5.19e4 - 3.77e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-1.48e4 + 4.55e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (6.62e4 - 4.81e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (1.56e4 + 1.13e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + 5.61e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-5.86e4 + 4.26e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−16.31148800233107032263224745031, −15.38734875316717829953607261499, −14.19041436532279985656183129986, −12.73886970446292995777407065854, −10.80246089527001504911901978095, −9.586087884755846087126450944971, −8.769320265758235392043465715120, −7.06154981913704227247822627425, −5.61359959192173653754190112518, −2.85151171126932428844207657732,
0.66898356107688537021889454871, 2.25963384495601108280514847871, 5.28022839312774204514443302873, 7.30426829341206810540739618067, 8.909946309633192176411221040153, 9.877280785442958100519787083009, 10.99595652769317884240656430964, 12.80167634448011821183290093619, 13.48278155830654584093253508102, 15.02885069910665372278345403836
