L(s) = 1 | + (0.370 − 1.69i)3-s + 5-s + 0.0825i·7-s + (−2.72 − 1.25i)9-s − 1.67·11-s + 4.50i·13-s + (0.370 − 1.69i)15-s + 1.74i·17-s − 1.02·19-s + (0.139 + 0.0305i)21-s + 3.31i·23-s + 25-s + (−3.13 + 4.14i)27-s − 3.47i·29-s − 4.73i·31-s + ⋯ |
L(s) = 1 | + (0.213 − 0.976i)3-s + 0.447·5-s + 0.0312i·7-s + (−0.908 − 0.417i)9-s − 0.503·11-s + 1.24i·13-s + (0.0956 − 0.436i)15-s + 0.422i·17-s − 0.234·19-s + (0.0304 + 0.00667i)21-s + 0.692i·23-s + 0.200·25-s + (−0.602 + 0.798i)27-s − 0.645i·29-s − 0.850i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.657379984\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.657379984\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.370 + 1.69i)T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (5.31 + 6.22i)T \) |
good | 7 | \( 1 - 0.0825iT - 7T^{2} \) |
| 11 | \( 1 + 1.67T + 11T^{2} \) |
| 13 | \( 1 - 4.50iT - 13T^{2} \) |
| 17 | \( 1 - 1.74iT - 17T^{2} \) |
| 19 | \( 1 + 1.02T + 19T^{2} \) |
| 23 | \( 1 - 3.31iT - 23T^{2} \) |
| 29 | \( 1 + 3.47iT - 29T^{2} \) |
| 31 | \( 1 + 4.73iT - 31T^{2} \) |
| 37 | \( 1 - 1.03T + 37T^{2} \) |
| 41 | \( 1 - 4.96T + 41T^{2} \) |
| 43 | \( 1 - 6.28iT - 43T^{2} \) |
| 47 | \( 1 - 12.7iT - 47T^{2} \) |
| 53 | \( 1 + 1.52T + 53T^{2} \) |
| 59 | \( 1 - 7.15iT - 59T^{2} \) |
| 61 | \( 1 - 1.24iT - 61T^{2} \) |
| 71 | \( 1 - 8.32iT - 71T^{2} \) |
| 73 | \( 1 - 9.54T + 73T^{2} \) |
| 79 | \( 1 - 4.58iT - 79T^{2} \) |
| 83 | \( 1 - 2.48iT - 83T^{2} \) |
| 89 | \( 1 - 2.00iT - 89T^{2} \) |
| 97 | \( 1 + 8.09iT - 97T^{2} \) |
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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
−8.405960334600936557267800461626, −7.69448922390273510461787416761, −7.13805820863841706932467087660, −6.13402606008174534296193323453, −5.94830905103764556004471098857, −4.75290939308258643680188353127, −3.87386661506927251939015546616, −2.71622525589024582409313035693, −2.07023899256166429551575837441, −1.10447241474000083808396902225,
0.48759801156173205395921222125, 2.15842652302389622022901336599, 2.96757103761359528306620890460, 3.68751624740381594991323648236, 4.76616775296581480189275467579, 5.28977805254941460141059511683, 5.94047870679524040188232522532, 6.94488784604689906374475653927, 7.81478650166990427165150404478, 8.541984539177977882460219351977