| L(s) = 1 | − 4·7-s + 11-s + 13-s − 7·17-s + 3·19-s + 4·29-s − 6·31-s + 8·37-s + 5·41-s − 4·43-s − 12·47-s + 9·49-s − 10·53-s + 4·59-s + 8·61-s − 9·67-s − 8·71-s − 13·73-s − 4·77-s − 8·79-s − 3·83-s + 11·89-s − 4·91-s + 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 0.301·11-s + 0.277·13-s − 1.69·17-s + 0.688·19-s + 0.742·29-s − 1.07·31-s + 1.31·37-s + 0.780·41-s − 0.609·43-s − 1.75·47-s + 9/7·49-s − 1.37·53-s + 0.520·59-s + 1.02·61-s − 1.09·67-s − 0.949·71-s − 1.52·73-s − 0.455·77-s − 0.900·79-s − 0.329·83-s + 1.16·89-s − 0.419·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8811398515\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8811398515\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.69265852853776, −14.04718325638401, −13.31283864505791, −13.10221620603634, −12.80380443062726, −11.97193323832595, −11.50567356357692, −11.03147461811730, −10.37802562406504, −9.837021389823479, −9.327851728977370, −9.008825704613243, −8.340501893372112, −7.666449424852500, −6.911636490554824, −6.643948593277134, −6.072082832643742, −5.558493197678717, −4.583747133212224, −4.244947267879478, −3.340641847997155, −3.016975158337125, −2.219652613316736, −1.360549925640037, −0.3359802759252648,
0.3359802759252648, 1.360549925640037, 2.219652613316736, 3.016975158337125, 3.340641847997155, 4.244947267879478, 4.583747133212224, 5.558493197678717, 6.072082832643742, 6.643948593277134, 6.911636490554824, 7.666449424852500, 8.340501893372112, 9.008825704613243, 9.327851728977370, 9.837021389823479, 10.37802562406504, 11.03147461811730, 11.50567356357692, 11.97193323832595, 12.80380443062726, 13.10221620603634, 13.31283864505791, 14.04718325638401, 14.69265852853776
