| L(s) = 1 | + (−0.882 + 2.71i)2-s + (4.90 − 1.71i)3-s + (−0.124 − 0.0905i)4-s + (4.91 − 1.59i)5-s + (0.325 + 14.8i)6-s + (−9.45 + 13.0i)7-s + (−18.1 + 13.1i)8-s + (21.1 − 16.8i)9-s + 14.7i·10-s + (−5.09 − 36.1i)11-s + (−0.766 − 0.230i)12-s + (16.5 + 5.38i)13-s + (−26.9 − 37.1i)14-s + (21.3 − 16.2i)15-s + (−20.1 − 62.0i)16-s + (−30.8 − 94.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.311 + 0.960i)2-s + (0.944 − 0.329i)3-s + (−0.0155 − 0.0113i)4-s + (0.439 − 0.142i)5-s + (0.0221 + 1.00i)6-s + (−0.510 + 0.702i)7-s + (−0.801 + 0.581i)8-s + (0.782 − 0.622i)9-s + 0.466i·10-s + (−0.139 − 0.990i)11-s + (−0.0184 − 0.00554i)12-s + (0.353 + 0.114i)13-s + (−0.515 − 0.709i)14-s + (0.368 − 0.279i)15-s + (−0.314 − 0.969i)16-s + (−0.439 − 1.35i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.23933 + 0.745153i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23933 + 0.745153i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.90 + 1.71i)T \) |
| 11 | \( 1 + (5.09 + 36.1i)T \) |
| good | 2 | \( 1 + (0.882 - 2.71i)T + (-6.47 - 4.70i)T^{2} \) |
| 5 | \( 1 + (-4.91 + 1.59i)T + (101. - 73.4i)T^{2} \) |
| 7 | \( 1 + (9.45 - 13.0i)T + (-105. - 326. i)T^{2} \) |
| 13 | \( 1 + (-16.5 - 5.38i)T + (1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (30.8 + 94.8i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (27.1 + 37.3i)T + (-2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 - 113. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-110. - 80.2i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-26.9 + 82.9i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (216. + 157. i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-21.4 + 15.5i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 - 487. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-35.3 - 48.6i)T + (-3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-472. - 153. i)T + (1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (432. - 594. i)T + (-6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-817. + 265. i)T + (1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 - 507.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (980. - 318. i)T + (2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (173. - 238. i)T + (-1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (576. + 187. i)T + (3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-39.9 - 122. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + 811. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (-69.6 + 214. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−16.08845653053080266510311356188, −15.53526069369978445723398630798, −14.15524596708766301235918862222, −13.17101724773071223903801633166, −11.63296376826875181684806503581, −9.377999972402967986415012163626, −8.611481372456702636288596885177, −7.19650692967345661802187891516, −5.86777798127923540076194789436, −2.84341583733368694235431141278,
2.09925376066474982606005748051, 3.88069030332344983264937873919, 6.67402340235424028372737839532, 8.581826217731287210083299762417, 10.17051398608881263470064274947, 10.34466981954524859079962270063, 12.39235967277140646818414853011, 13.46130146427127657645952900619, 14.81736892621768921957645459917, 15.82520053480503653290614241292
