| L(s) = 1 | + 3-s + 2·7-s + 9-s + 13-s + 6·19-s + 2·21-s − 2·23-s + 27-s − 4·29-s − 8·31-s + 10·37-s + 39-s − 2·41-s − 12·43-s − 4·47-s − 3·49-s − 6·53-s + 6·57-s − 8·59-s − 2·61-s + 2·63-s − 4·67-s − 2·69-s + 81-s + 4·83-s − 4·87-s − 14·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.277·13-s + 1.37·19-s + 0.436·21-s − 0.417·23-s + 0.192·27-s − 0.742·29-s − 1.43·31-s + 1.64·37-s + 0.160·39-s − 0.312·41-s − 1.82·43-s − 0.583·47-s − 3/7·49-s − 0.824·53-s + 0.794·57-s − 1.04·59-s − 0.256·61-s + 0.251·63-s − 0.488·67-s − 0.240·69-s + 1/9·81-s + 0.439·83-s − 0.428·87-s − 1.48·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−15.11588179579724, −14.86893372364635, −14.30676876789499, −13.85023110483570, −13.25920925451096, −12.88040761986451, −12.13099221402284, −11.52560053473998, −11.21574636007819, −10.56689893335294, −9.807878635435400, −9.453261218233965, −8.891820142446819, −8.146191386011124, −7.802643870583921, −7.321251496583105, −6.560461636191191, −5.886173853222416, −5.183910814141110, −4.727769701046266, −3.881649568960806, −3.362886164611676, −2.665677615431609, −1.711935071510777, −1.337742073512650, 0,
1.337742073512650, 1.711935071510777, 2.665677615431609, 3.362886164611676, 3.881649568960806, 4.727769701046266, 5.183910814141110, 5.886173853222416, 6.560461636191191, 7.321251496583105, 7.802643870583921, 8.146191386011124, 8.891820142446819, 9.453261218233965, 9.807878635435400, 10.56689893335294, 11.21574636007819, 11.52560053473998, 12.13099221402284, 12.88040761986451, 13.25920925451096, 13.85023110483570, 14.30676876789499, 14.86893372364635, 15.11588179579724
