| L(s) = 1 | + (−0.184 + 0.253i)2-s + (0.535 − 1.64i)3-s + (1.20 + 3.71i)4-s + (5.99 − 4.35i)5-s + (0.318 + 0.438i)6-s + (−9.53 + 3.09i)7-s + (−2.35 − 0.764i)8-s + (−2.42 − 1.76i)9-s + 2.32i·10-s + (−10.2 + 3.93i)11-s + 6.75·12-s + (2.00 − 2.75i)13-s + (0.970 − 2.98i)14-s + (−3.96 − 12.2i)15-s + (−12.0 + 8.71i)16-s + (−9.14 − 12.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.0920 + 0.126i)2-s + (0.178 − 0.549i)3-s + (0.301 + 0.927i)4-s + (1.19 − 0.871i)5-s + (0.0531 + 0.0731i)6-s + (−1.36 + 0.442i)7-s + (−0.294 − 0.0955i)8-s + (−0.269 − 0.195i)9-s + 0.232i·10-s + (−0.933 + 0.358i)11-s + 0.563·12-s + (0.153 − 0.211i)13-s + (0.0692 − 0.213i)14-s + (−0.264 − 0.814i)15-s + (−0.750 + 0.544i)16-s + (−0.537 − 0.740i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0174i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0174i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.07879 - 0.00938772i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07879 - 0.00938772i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.535 + 1.64i)T \) |
| 11 | \( 1 + (10.2 - 3.93i)T \) |
| good | 2 | \( 1 + (0.184 - 0.253i)T + (-1.23 - 3.80i)T^{2} \) |
| 5 | \( 1 + (-5.99 + 4.35i)T + (7.72 - 23.7i)T^{2} \) |
| 7 | \( 1 + (9.53 - 3.09i)T + (39.6 - 28.8i)T^{2} \) |
| 13 | \( 1 + (-2.00 + 2.75i)T + (-52.2 - 160. i)T^{2} \) |
| 17 | \( 1 + (9.14 + 12.5i)T + (-89.3 + 274. i)T^{2} \) |
| 19 | \( 1 + (-29.1 - 9.46i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 - 7.67T + 529T^{2} \) |
| 29 | \( 1 + (-3.21 + 1.04i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (-3.27 - 2.37i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (0.734 + 2.25i)T + (-1.10e3 + 804. i)T^{2} \) |
| 41 | \( 1 + (-6.69 - 2.17i)T + (1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + 3.99iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-15.2 + 47.0i)T + (-1.78e3 - 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-48.3 - 35.1i)T + (868. + 2.67e3i)T^{2} \) |
| 59 | \( 1 + (3.39 + 10.4i)T + (-2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-43.6 - 60.1i)T + (-1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + 3.22T + 4.48e3T^{2} \) |
| 71 | \( 1 + (94.5 - 68.7i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-17.8 + 5.78i)T + (4.31e3 - 3.13e3i)T^{2} \) |
| 79 | \( 1 + (2.06 - 2.84i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (86.4 + 119. i)T + (-2.12e3 + 6.55e3i)T^{2} \) |
| 89 | \( 1 + 65.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + (52.4 + 38.1i)T + (2.90e3 + 8.94e3i)T^{2} \) |
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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
−16.48439531518953433049746996693, −15.73612441986101390246673765710, −13.53454988467521956418465371740, −13.00645820561289224175521406267, −12.05088044442935858022597532266, −9.820611425739183496023579811053, −8.782208693993455986944533521542, −7.16882412832442390930857642734, −5.66684548670637219790598277779, −2.75245676862392108387586191427,
2.85141994790067530153574085742, 5.65463462597109520945196575610, 6.77404040014754064869687156632, 9.414849626623867607636312266442, 10.14617726852584079758915368421, 10.98034822453394853065957318825, 13.29455306748765307806619509521, 14.06444709357203262073258886312, 15.37130443916638479461556309146, 16.27617110536437385927014647174
