L(s) = 1 | + 2·5-s + 2·13-s + 6·17-s + 4·19-s + 4·23-s − 25-s − 6·29-s + 8·31-s − 10·37-s − 10·41-s + 12·43-s − 8·47-s − 6·53-s + 4·59-s + 10·61-s + 4·65-s + 12·67-s − 4·71-s − 2·73-s + 8·79-s + 4·83-s + 12·85-s + 6·89-s + 8·95-s − 10·97-s + 10·101-s − 16·107-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 0.554·13-s + 1.45·17-s + 0.917·19-s + 0.834·23-s − 1/5·25-s − 1.11·29-s + 1.43·31-s − 1.64·37-s − 1.56·41-s + 1.82·43-s − 1.16·47-s − 0.824·53-s + 0.520·59-s + 1.28·61-s + 0.496·65-s + 1.46·67-s − 0.474·71-s − 0.234·73-s + 0.900·79-s + 0.439·83-s + 1.30·85-s + 0.635·89-s + 0.820·95-s − 1.01·97-s + 0.995·101-s − 1.54·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.533531301\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.533531301\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.577574048820412073979821096434, −7.86004275519657164377594722946, −7.05267897803070418923661148552, −6.27830454481138490924310987858, −5.47919414246685677243404259713, −5.06048784569425782183212035239, −3.72769041359201298820498364558, −3.08157339545259112628074418683, −1.91473513602758200866610556345, −1.00371826659586316798708144945,
1.00371826659586316798708144945, 1.91473513602758200866610556345, 3.08157339545259112628074418683, 3.72769041359201298820498364558, 5.06048784569425782183212035239, 5.47919414246685677243404259713, 6.27830454481138490924310987858, 7.05267897803070418923661148552, 7.86004275519657164377594722946, 8.577574048820412073979821096434