| L(s) = 1 | + (−7.14 + 4.12i)2-s + (−15.5 − 0.743i)3-s + (18.0 − 31.1i)4-s + (3.57 + 6.18i)5-s + (114. − 58.8i)6-s + (122.5 − 42.4i)7-s + 32.9i·8-s + (241. + 23.1i)9-s + (−51 − 29.4i)10-s + (−389. − 224. i)11-s + (−303. + 472. i)12-s − 523. i·13-s + (−699. + 808. i)14-s + (−51 − 98.9i)15-s + (439. + 762. i)16-s + (592. − 1.02e3i)17-s + ⋯ |
| L(s) = 1 | + (−1.26 + 0.728i)2-s + (−0.998 − 0.0476i)3-s + (0.562 − 0.974i)4-s + (0.0638 + 0.110i)5-s + (1.29 − 0.667i)6-s + (0.944 − 0.327i)7-s + 0.182i·8-s + (0.995 + 0.0952i)9-s + (−0.161 − 0.0931i)10-s + (−0.969 − 0.559i)11-s + (−0.608 + 0.946i)12-s − 0.858i·13-s + (−0.954 + 1.10i)14-s + (−0.0585 − 0.113i)15-s + (0.429 + 0.744i)16-s + (0.497 − 0.861i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.450802 - 0.148064i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.450802 - 0.148064i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (15.5 + 0.743i)T \) |
| 7 | \( 1 + (-122.5 + 42.4i)T \) |
| good | 2 | \( 1 + (7.14 - 4.12i)T + (16 - 27.7i)T^{2} \) |
| 5 | \( 1 + (-3.57 - 6.18i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 11 | \( 1 + (389. + 224. i)T + (8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + 523. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + (-592. + 1.02e3i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (471 - 271. i)T + (1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-3.29e3 + 1.90e3i)T + (3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 - 2.62e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + (7.51e3 + 4.33e3i)T + (1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-3.05e3 - 5.28e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 - 199.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.74e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.30e4 + 2.26e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-1.19e4 - 6.87e3i)T + (2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-1.45e4 + 2.51e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (3.85e3 - 2.22e3i)T + (4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (691 - 1.19e3i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 8.32e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-2.81e4 - 1.62e4i)T + (1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (1.05e4 + 1.82e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + 3.66e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-3.07e4 - 5.32e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + 1.20e5iT - 8.58e9T^{2} \) |
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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
−17.01707867818977961322080250100, −16.25839888568219250817762694888, −14.95045607192638373432226655400, −12.94887967197967864913431113742, −11.08882486414568411187771521703, −10.18743895073344684522437725294, −8.288909663416830269219981765552, −7.08622777285103467891577035334, −5.32968833404159347265623983130, −0.63303567238911436511624157476,
1.59860396022289990483488352929, 5.17027613606820683888448949498, 7.54332496797997857707927538325, 9.184312345225497774560397706969, 10.61724274032165296305139180837, 11.41481573444631667480996361947, 12.71718064443127449459048683110, 14.96707493322973232095708603254, 16.60432493672130650841759171383, 17.53200032326848587507536260141
