| L(s) = 1 | + (3.38 + 10.4i)2-s + (−7.28 − 5.29i)3-s + (−71.3 + 51.8i)4-s + (8.39 − 25.8i)5-s + (30.4 − 93.8i)6-s + (−110. + 80.1i)7-s + (−497. − 361. i)8-s + (25.0 + 77.0i)9-s + 297.·10-s + (272. + 294. i)11-s + 793.·12-s + (32.7 + 100. i)13-s + (−1.20e3 − 878. i)14-s + (−197. + 143. i)15-s + (1.21e3 − 3.73e3i)16-s + (−583. + 1.79e3i)17-s + ⋯ |
| L(s) = 1 | + (0.598 + 1.84i)2-s + (−0.467 − 0.339i)3-s + (−2.22 + 1.61i)4-s + (0.150 − 0.462i)5-s + (0.345 − 1.06i)6-s + (−0.851 + 0.618i)7-s + (−2.75 − 1.99i)8-s + (0.103 + 0.317i)9-s + 0.941·10-s + (0.678 + 0.734i)11-s + 1.59·12-s + (0.0536 + 0.165i)13-s + (−1.64 − 1.19i)14-s + (−0.227 + 0.164i)15-s + (1.18 − 3.64i)16-s + (−0.490 + 1.50i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.351920 - 0.964006i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.351920 - 0.964006i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (7.28 + 5.29i)T \) |
| 11 | \( 1 + (-272. - 294. i)T \) |
| good | 2 | \( 1 + (-3.38 - 10.4i)T + (-25.8 + 18.8i)T^{2} \) |
| 5 | \( 1 + (-8.39 + 25.8i)T + (-2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (110. - 80.1i)T + (5.19e3 - 1.59e4i)T^{2} \) |
| 13 | \( 1 + (-32.7 - 100. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (583. - 1.79e3i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (8.60 + 6.25i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + 3.87e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (-3.41e3 + 2.47e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-1.25e3 - 3.84e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-5.13e3 + 3.72e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-6.89e3 - 5.01e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 - 1.45e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (7.37e3 + 5.36e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (6.21e3 + 1.91e4i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (2.87e4 - 2.08e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-9.01e3 + 2.77e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 - 1.04e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (5.27e3 - 1.62e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (1.58e4 - 1.15e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-1.47e4 - 4.53e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (3.02e4 - 9.30e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + 1.46e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (2.50e4 + 7.70e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−16.29094996369134768471419351298, −15.41235620935863993210643291588, −14.23126493622604958189196938587, −12.88935313958584108446297075448, −12.33819609674515810833438884817, −9.500427734275721495828853287952, −8.257599723518042031816195205671, −6.67336343824055461852058813350, −5.87057969752689419366013480577, −4.26789790455002863376801051277,
0.55592450852828402904740890920, 2.96989050506120901142141026358, 4.33550676984534003605482329666, 6.15310800610373012124984584982, 9.276298679508904175403082457767, 10.24552385429167451132021911919, 11.21024631674454256903501116345, 12.23037386723700454825571579166, 13.55558992836801407363324436604, 14.30091708159437269773060750453
