L(s) = 1 | + (−3.08 − 2.24i)2-s + (−2.78 + 8.55i)3-s + (−5.38 − 16.5i)4-s + (3.77 − 2.74i)5-s + (27.7 − 20.1i)6-s + (49.4 + 152. i)7-s + (−58.2 + 179. i)8-s + (−65.5 − 47.6i)9-s − 17.8·10-s + (85.6 + 392. i)11-s + 156.·12-s + (821. + 597. i)13-s + (188. − 580. i)14-s + (12.9 + 39.9i)15-s + (131. − 95.2i)16-s + (−1.64e3 + 1.19e3i)17-s + ⋯ |
L(s) = 1 | + (−0.545 − 0.396i)2-s + (−0.178 + 0.549i)3-s + (−0.168 − 0.518i)4-s + (0.0675 − 0.0490i)5-s + (0.315 − 0.228i)6-s + (0.381 + 1.17i)7-s + (−0.322 + 0.991i)8-s + (−0.269 − 0.195i)9-s − 0.0563·10-s + (0.213 + 0.976i)11-s + 0.314·12-s + (1.34 + 0.979i)13-s + (0.257 − 0.791i)14-s + (0.0149 + 0.0458i)15-s + (0.128 − 0.0930i)16-s + (−1.38 + 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.753762 + 0.505412i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.753762 + 0.505412i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.78 - 8.55i)T \) |
| 11 | \( 1 + (-85.6 - 392. i)T \) |
good | 2 | \( 1 + (3.08 + 2.24i)T + (9.88 + 30.4i)T^{2} \) |
| 5 | \( 1 + (-3.77 + 2.74i)T + (965. - 2.97e3i)T^{2} \) |
| 7 | \( 1 + (-49.4 - 152. i)T + (-1.35e4 + 9.87e3i)T^{2} \) |
| 13 | \( 1 + (-821. - 597. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (1.64e3 - 1.19e3i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-919. + 2.83e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + 1.90e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (487. + 1.49e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-1.91e3 - 1.38e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-1.23e3 - 3.78e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-1.06e3 + 3.27e3i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 + 2.63e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (5.00e3 - 1.54e4i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-1.78e4 - 1.29e4i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (9.00e3 + 2.77e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (3.58e3 - 2.60e3i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + 2.80e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-4.87e3 + 3.53e3i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (1.47e4 + 4.54e4i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-4.31e4 - 3.13e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-6.51e4 + 4.73e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 - 3.79e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-7.80e4 - 5.67e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−15.64739730139934341206895558023, −15.05524235746435906958380129787, −13.55842705240138682342167758004, −11.72291310817173242414934258645, −10.93644230288368689683955593638, −9.368557094588458882661189018354, −8.752388006286249467030117867174, −6.20354988144334638009597437460, −4.67080084356642023393893983262, −1.92835404459372927894442462885,
0.70479447530042591513967358242, 3.74642703013542517352310629479, 6.22109953743811717202580678050, 7.65033592372723297336708114673, 8.544575593165690028698657120451, 10.37551735914806131806147948632, 11.70100918688690162165040729693, 13.26928752607078750389779282769, 13.93803524854912541677189557126, 15.97420442817110779053339166137