L(s) = 1 | + (0.259 + 1.98i)2-s + (−2.58 + 1.49i)3-s + (−3.86 + 1.02i)4-s + (−3.40 + 5.89i)5-s + (−3.62 − 4.73i)6-s + (−3.04 − 7.39i)8-s + (−0.0457 + 0.0792i)9-s + (−12.5 − 5.21i)10-s + (12.4 − 7.17i)11-s + (8.45 − 8.42i)12-s − 4.62·13-s − 20.3i·15-s + (13.8 − 7.95i)16-s + (5.80 + 10.0i)17-s + (−0.168 − 0.0701i)18-s + (−19.4 − 11.2i)19-s + ⋯ |
L(s) = 1 | + (0.129 + 0.991i)2-s + (−0.861 + 0.497i)3-s + (−0.966 + 0.257i)4-s + (−0.680 + 1.17i)5-s + (−0.604 − 0.789i)6-s + (−0.380 − 0.924i)8-s + (−0.00508 + 0.00880i)9-s + (−1.25 − 0.521i)10-s + (1.12 − 0.651i)11-s + (0.704 − 0.702i)12-s − 0.355·13-s − 1.35i·15-s + (0.867 − 0.496i)16-s + (0.341 + 0.591i)17-s + (−0.00938 − 0.00389i)18-s + (−1.02 − 0.590i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0100 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0100 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.232539 - 0.234887i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.232539 - 0.234887i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.259 - 1.98i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (2.58 - 1.49i)T + (4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (3.40 - 5.89i)T + (-12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-12.4 + 7.17i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 4.62T + 169T^{2} \) |
| 17 | \( 1 + (-5.80 - 10.0i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (19.4 + 11.2i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-0.739 - 0.426i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 42.8T + 841T^{2} \) |
| 31 | \( 1 + (1.34 - 0.774i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (22.5 - 38.9i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 36.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 27.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (36.5 + 21.0i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (21.7 + 37.5i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (46.7 - 26.9i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (7.97 - 13.8i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (79.6 - 45.9i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 16.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-4.79 - 8.30i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (49.3 + 28.4i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 14.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-50.0 + 86.7i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 68.2T + 9.40e3T^{2} \) |
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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
−13.10664934396372708436480992861, −11.80788228763925629217902212982, −11.09898825911558775283091360452, −10.15733490721954642872599819951, −8.872303901448528283575719403544, −7.70684979228507735516033839031, −6.64626871598520277448700311498, −5.90973488745466191120722845875, −4.53075554123609177020578434419, −3.45723138965212881566370065257,
0.21292756149006271899920101026, 1.58069092327661275841665264058, 3.83736759478703208251734075247, 4.82417075878586233468904039335, 5.97259185711137205346496247939, 7.48224360158096012655426044940, 8.825369109446719596373507262666, 9.521184212288579425143253002759, 10.95146222085559687020292284728, 11.77959108842396672148521581380