| L(s) = 1 | + (−1.65 + 0.578i)2-s + (−0.727 + 0.579i)4-s + (0.825 − 1.71i)5-s + (−1.24 + 1.56i)7-s + (4.59 − 7.31i)8-s + (−0.373 + 3.31i)10-s + (6.63 + 10.5i)11-s + (−3.80 − 0.868i)13-s + (1.15 − 3.30i)14-s + (−2.54 + 11.1i)16-s + (−7.59 + 7.59i)17-s + (−0.137 − 0.0154i)19-s + (0.393 + 1.72i)20-s + (−17.0 − 13.6i)22-s + (−26.7 + 12.8i)23-s + ⋯ |
| L(s) = 1 | + (−0.826 + 0.289i)2-s + (−0.181 + 0.144i)4-s + (0.165 − 0.342i)5-s + (−0.178 + 0.223i)7-s + (0.574 − 0.914i)8-s + (−0.0373 + 0.331i)10-s + (0.602 + 0.959i)11-s + (−0.292 − 0.0668i)13-s + (0.0827 − 0.236i)14-s + (−0.158 + 0.695i)16-s + (−0.446 + 0.446i)17-s + (−0.00723 − 0.000815i)19-s + (0.0196 + 0.0862i)20-s + (−0.775 − 0.618i)22-s + (−1.16 + 0.559i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 - 0.600i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.163601 + 0.490326i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.163601 + 0.490326i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 + (8.15 + 27.8i)T \) |
| good | 2 | \( 1 + (1.65 - 0.578i)T + (3.12 - 2.49i)T^{2} \) |
| 5 | \( 1 + (-0.825 + 1.71i)T + (-15.5 - 19.5i)T^{2} \) |
| 7 | \( 1 + (1.24 - 1.56i)T + (-10.9 - 47.7i)T^{2} \) |
| 11 | \( 1 + (-6.63 - 10.5i)T + (-52.4 + 109. i)T^{2} \) |
| 13 | \( 1 + (3.80 + 0.868i)T + (152. + 73.3i)T^{2} \) |
| 17 | \( 1 + (7.59 - 7.59i)T - 289iT^{2} \) |
| 19 | \( 1 + (0.137 + 0.0154i)T + (351. + 80.3i)T^{2} \) |
| 23 | \( 1 + (26.7 - 12.8i)T + (329. - 413. i)T^{2} \) |
| 31 | \( 1 + (54.1 - 18.9i)T + (751. - 599. i)T^{2} \) |
| 37 | \( 1 + (29.9 - 47.6i)T + (-593. - 1.23e3i)T^{2} \) |
| 41 | \( 1 + (-25.9 - 25.9i)T + 1.68e3iT^{2} \) |
| 43 | \( 1 + (-5.16 + 14.7i)T + (-1.44e3 - 1.15e3i)T^{2} \) |
| 47 | \( 1 + (55.9 - 35.1i)T + (958. - 1.99e3i)T^{2} \) |
| 53 | \( 1 + (29.3 + 14.1i)T + (1.75e3 + 2.19e3i)T^{2} \) |
| 59 | \( 1 + 0.396T + 3.48e3T^{2} \) |
| 61 | \( 1 + (6.99 + 62.1i)T + (-3.62e3 + 828. i)T^{2} \) |
| 67 | \( 1 + (21.7 - 4.97i)T + (4.04e3 - 1.94e3i)T^{2} \) |
| 71 | \( 1 + (-64.8 - 14.7i)T + (4.54e3 + 2.18e3i)T^{2} \) |
| 73 | \( 1 + (-76.3 - 26.7i)T + (4.16e3 + 3.32e3i)T^{2} \) |
| 79 | \( 1 + (-67.7 - 42.5i)T + (2.70e3 + 5.62e3i)T^{2} \) |
| 83 | \( 1 + (-68.4 - 85.8i)T + (-1.53e3 + 6.71e3i)T^{2} \) |
| 89 | \( 1 + (-77.7 + 27.2i)T + (6.19e3 - 4.93e3i)T^{2} \) |
| 97 | \( 1 + (-19.5 + 173. i)T + (-9.17e3 - 2.09e3i)T^{2} \) |
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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
−12.31100292155733442963824780073, −11.02309548855391447874729182644, −9.746222782734618290061733280099, −9.389448760592152735681751072934, −8.324900246591633533665125384203, −7.40981905836843885103699130683, −6.39419279205575817849275625015, −4.90466433375562160158121779721, −3.73693222402105637700073996828, −1.67622989339904857142524565956,
0.35373241627597734049600468085, 2.09658040806576725334340134837, 3.78608688430552067459137117095, 5.24909609103754339899875423635, 6.44574087167285998644682062040, 7.62740454418973226658834661348, 8.775953299225926819892404365602, 9.368596926452578344764325629453, 10.49208123403816152663278871622, 11.00426265866567836813512076343
