| L(s) = 1 | + (1.63 + 2.59i)2-s + (−2.35 + 4.88i)4-s + (4.43 + 1.01i)5-s + (10.7 − 5.18i)7-s + (−4.32 + 0.487i)8-s + (4.61 + 13.1i)10-s + (−2.90 − 0.326i)11-s + (2.24 − 1.79i)13-s + (31.0 + 19.5i)14-s + (5.18 + 6.49i)16-s + (−2.44 + 2.44i)17-s + (−31.9 + 11.1i)19-s + (−15.3 + 19.2i)20-s + (−3.88 − 8.07i)22-s + (−9.24 − 40.5i)23-s + ⋯ |
| L(s) = 1 | + (0.816 + 1.29i)2-s + (−0.587 + 1.22i)4-s + (0.887 + 0.202i)5-s + (1.53 − 0.740i)7-s + (−0.540 + 0.0609i)8-s + (0.461 + 1.31i)10-s + (−0.263 − 0.0297i)11-s + (0.173 − 0.137i)13-s + (2.21 + 1.39i)14-s + (0.323 + 0.406i)16-s + (−0.144 + 0.144i)17-s + (−1.68 + 0.588i)19-s + (−0.769 + 0.964i)20-s + (−0.176 − 0.366i)22-s + (−0.402 − 1.76i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0648 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0648 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.08485 + 2.22470i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.08485 + 2.22470i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 + (18.6 - 22.1i)T \) |
| good | 2 | \( 1 + (-1.63 - 2.59i)T + (-1.73 + 3.60i)T^{2} \) |
| 5 | \( 1 + (-4.43 - 1.01i)T + (22.5 + 10.8i)T^{2} \) |
| 7 | \( 1 + (-10.7 + 5.18i)T + (30.5 - 38.3i)T^{2} \) |
| 11 | \( 1 + (2.90 + 0.326i)T + (117. + 26.9i)T^{2} \) |
| 13 | \( 1 + (-2.24 + 1.79i)T + (37.6 - 164. i)T^{2} \) |
| 17 | \( 1 + (2.44 - 2.44i)T - 289iT^{2} \) |
| 19 | \( 1 + (31.9 - 11.1i)T + (282. - 225. i)T^{2} \) |
| 23 | \( 1 + (9.24 + 40.5i)T + (-476. + 229. i)T^{2} \) |
| 31 | \( 1 + (-7.40 - 11.7i)T + (-416. + 865. i)T^{2} \) |
| 37 | \( 1 + (19.6 - 2.21i)T + (1.33e3 - 304. i)T^{2} \) |
| 41 | \( 1 + (-15.2 - 15.2i)T + 1.68e3iT^{2} \) |
| 43 | \( 1 + (-1.58 - 0.996i)T + (802. + 1.66e3i)T^{2} \) |
| 47 | \( 1 + (-0.858 + 7.61i)T + (-2.15e3 - 491. i)T^{2} \) |
| 53 | \( 1 + (-7.13 + 31.2i)T + (-2.53e3 - 1.21e3i)T^{2} \) |
| 59 | \( 1 - 9.47T + 3.48e3T^{2} \) |
| 61 | \( 1 + (19.5 - 55.9i)T + (-2.90e3 - 2.32e3i)T^{2} \) |
| 67 | \( 1 + (45.3 + 36.1i)T + (998. + 4.37e3i)T^{2} \) |
| 71 | \( 1 + (-61.2 + 48.8i)T + (1.12e3 - 4.91e3i)T^{2} \) |
| 73 | \( 1 + (-40.7 + 64.8i)T + (-2.31e3 - 4.80e3i)T^{2} \) |
| 79 | \( 1 + (-10.0 - 89.0i)T + (-6.08e3 + 1.38e3i)T^{2} \) |
| 83 | \( 1 + (42.5 + 20.4i)T + (4.29e3 + 5.38e3i)T^{2} \) |
| 89 | \( 1 + (-31.5 - 50.2i)T + (-3.43e3 + 7.13e3i)T^{2} \) |
| 97 | \( 1 + (41.8 + 119. i)T + (-7.35e3 + 5.86e3i)T^{2} \) |
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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
−12.39595033603462685825841451042, −10.82113401210467755592240515042, −10.38521345954265518586833897080, −8.575507162585676480465399521931, −7.965285579205672355139193117440, −6.82055650371444344127488032077, −5.98578392607267570925834189314, −4.91930973259582578598642656793, −4.10854235768937376273528453812, −1.90982940106309203151152164798,
1.69555688287061144396905382783, 2.34190406427340968614934777042, 4.09918749164545477369985064436, 5.11660867989969669809664393246, 5.86977462816311988146116621688, 7.73467546632047408520864768238, 8.892432063055210314104403637202, 9.859758270672257480137156991760, 10.98270872834117906926221180035, 11.44891985497731571674863398652
