| L(s) = 1 | + (4.95 − 80.8i)3-s − 404. i·5-s − 262.·7-s + (−6.51e3 − 801. i)9-s + 6.26e3i·11-s + 4.39e4·13-s + (−3.26e4 − 2.00e3i)15-s + 9.11e4i·17-s + 1.70e5·19-s + (−1.29e3 + 2.12e4i)21-s − 2.34e5i·23-s + 2.27e5·25-s + (−9.70e4 + 5.22e5i)27-s − 1.23e6i·29-s + 1.50e6·31-s + ⋯ |
| L(s) = 1 | + (0.0611 − 0.998i)3-s − 0.646i·5-s − 0.109·7-s + (−0.992 − 0.122i)9-s + 0.428i·11-s + 1.53·13-s + (−0.645 − 0.0395i)15-s + 1.09i·17-s + 1.30·19-s + (−0.00668 + 0.109i)21-s − 0.839i·23-s + 0.581·25-s + (−0.182 + 0.983i)27-s − 1.74i·29-s + 1.63·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0611 + 0.998i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.0611 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.386456330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.386456330\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.95 + 80.8i)T \) |
| good | 5 | \( 1 + 404. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 262.T + 5.76e6T^{2} \) |
| 11 | \( 1 - 6.26e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 4.39e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 9.11e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.70e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 2.34e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 1.23e6iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.50e6T + 8.52e11T^{2} \) |
| 37 | \( 1 + 2.95e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 2.45e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 3.32e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 5.63e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 4.50e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 9.47e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 5.61e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 3.14e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 1.69e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 5.16e6T + 8.06e14T^{2} \) |
| 79 | \( 1 - 5.76e6T + 1.51e15T^{2} \) |
| 83 | \( 1 - 4.72e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 6.72e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.36e8T + 7.83e15T^{2} \) |
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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
−11.01239213603115318233872724115, −9.685788617555455594063338805222, −8.505142080864883361330068915524, −7.959562231946737482230293497045, −6.56874389018586320391514858315, −5.83082356181725312843501411668, −4.37298843495664771246517976839, −2.94577732498411155288119769580, −1.51254518646611571588208393361, −0.72264259613799179260935861227,
0.981735069571571063992396965212, 2.94654747067664074291169761160, 3.54749714097375071644222911302, 4.99112233692722565959178240936, 5.99115769342412587052367743984, 7.21450414259086104109554519773, 8.559134974104815321323569449101, 9.352538044941270594190570701148, 10.43577367702596323893551953323, 11.12621103732781107899641349175
