| L(s) = 1 | + (0.597 + 1.30i)2-s + (0.959 − 0.281i)3-s + (−0.0460 + 0.0531i)4-s + (−0.841 − 0.540i)5-s + (0.942 + 1.08i)6-s + (0.462 + 3.21i)7-s + (2.66 + 0.782i)8-s + (0.841 − 0.540i)9-s + (0.204 − 1.42i)10-s + (1.08 − 2.36i)11-s + (−0.0292 + 0.0639i)12-s + (0.116 − 0.810i)13-s + (−3.93 + 2.53i)14-s + (−0.959 − 0.281i)15-s + (0.588 + 4.09i)16-s + (0.275 + 0.317i)17-s + ⋯ |
| L(s) = 1 | + (0.422 + 0.925i)2-s + (0.553 − 0.162i)3-s + (−0.0230 + 0.0265i)4-s + (−0.376 − 0.241i)5-s + (0.384 + 0.443i)6-s + (0.174 + 1.21i)7-s + (0.941 + 0.276i)8-s + (0.280 − 0.180i)9-s + (0.0647 − 0.450i)10-s + (0.325 − 0.713i)11-s + (−0.00843 + 0.0184i)12-s + (0.0323 − 0.224i)13-s + (−1.05 + 0.676i)14-s + (−0.247 − 0.0727i)15-s + (0.147 + 1.02i)16-s + (0.0667 + 0.0769i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.82111 + 1.01503i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.82111 + 1.01503i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.959 + 0.281i)T \) |
| 5 | \( 1 + (0.841 + 0.540i)T \) |
| 23 | \( 1 + (-4.71 + 0.883i)T \) |
| good | 2 | \( 1 + (-0.597 - 1.30i)T + (-1.30 + 1.51i)T^{2} \) |
| 7 | \( 1 + (-0.462 - 3.21i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (-1.08 + 2.36i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.116 + 0.810i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-0.275 - 0.317i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (3.12 - 3.60i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (5.61 + 6.48i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (1.49 + 0.438i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (3.76 - 2.41i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (2.76 + 1.77i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (6.91 - 2.02i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + 4.01T + 47T^{2} \) |
| 53 | \( 1 + (0.974 + 6.78i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-1.42 + 9.89i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (6.66 + 1.95i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (2.21 + 4.84i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (-3.31 - 7.25i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-9.10 + 10.5i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (0.517 - 3.59i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (2.30 - 1.48i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (10.5 - 3.10i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-7.33 - 4.71i)T + (40.2 + 88.2i)T^{2} \) |
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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
−11.75427743493419721375943295887, −10.86588470833726962179736615775, −9.548371701603657926861599782660, −8.458157980109701647122946454282, −8.019082599223921800065944087711, −6.74337147146044470765105702586, −5.87153097137477023711146586182, −4.95326960751296883807557861613, −3.55393523620033374220816430836, −1.93693778716381144280867158052,
1.63003618014510787375984157629, 3.09870309496898527905566497955, 4.01431619507527962120995601539, 4.79911252902880604671891970415, 7.08205747677239048233461679900, 7.25260877538182966631253753090, 8.655242550329465445474156436049, 9.793943332960850880069274440122, 10.73287822848727564362865883682, 11.17749242707051959424771940624
