| L(s) = 1 | + 5·7-s − 3·11-s − 13-s + 3·17-s + 4·19-s − 6·23-s − 9·29-s − 5·31-s − 2·37-s + 2·43-s + 9·47-s + 18·49-s − 9·53-s − 9·59-s − 61-s + 5·67-s − 14·73-s − 15·77-s + 16·79-s + 15·83-s + 6·89-s − 5·91-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.88·7-s − 0.904·11-s − 0.277·13-s + 0.727·17-s + 0.917·19-s − 1.25·23-s − 1.67·29-s − 0.898·31-s − 0.328·37-s + 0.304·43-s + 1.31·47-s + 18/7·49-s − 1.23·53-s − 1.17·59-s − 0.128·61-s + 0.610·67-s − 1.63·73-s − 1.70·77-s + 1.80·79-s + 1.64·83-s + 0.635·89-s − 0.524·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.84094272237133, −14.38656413467373, −13.80851515021705, −13.56718649443414, −12.69024853514537, −12.17239216725043, −11.82699110363460, −11.15711510369822, −10.76150912970963, −10.38276930780595, −9.486726666829526, −9.226079859507106, −8.334539081390631, −7.861446677506565, −7.620776878980754, −7.161663767100237, −6.055971611198224, −5.474540901733376, −5.238576186438637, −4.550829702515688, −3.908768279210612, −3.223248273949020, −2.270385157760589, −1.842179447980613, −1.105813762797862, 0,
1.105813762797862, 1.842179447980613, 2.270385157760589, 3.223248273949020, 3.908768279210612, 4.550829702515688, 5.238576186438637, 5.474540901733376, 6.055971611198224, 7.161663767100237, 7.620776878980754, 7.861446677506565, 8.334539081390631, 9.226079859507106, 9.486726666829526, 10.38276930780595, 10.76150912970963, 11.15711510369822, 11.82699110363460, 12.17239216725043, 12.69024853514537, 13.56718649443414, 13.80851515021705, 14.38656413467373, 14.84094272237133
