| L(s) = 1 | + 2·3-s + 2·7-s + 2·9-s − 4·11-s + 4·21-s + 14·23-s + 6·27-s − 4·29-s + 4·31-s − 8·33-s + 4·37-s + 16·41-s + 2·43-s + 10·47-s − 6·49-s + 8·53-s − 8·59-s − 12·61-s + 4·63-s − 2·67-s + 28·69-s + 4·71-s − 8·77-s + 8·79-s + 11·81-s − 6·83-s − 8·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.755·7-s + 2/3·9-s − 1.20·11-s + 0.872·21-s + 2.91·23-s + 1.15·27-s − 0.742·29-s + 0.718·31-s − 1.39·33-s + 0.657·37-s + 2.49·41-s + 0.304·43-s + 1.45·47-s − 6/7·49-s + 1.09·53-s − 1.04·59-s − 1.53·61-s + 0.503·63-s − 0.244·67-s + 3.37·69-s + 0.474·71-s − 0.911·77-s + 0.900·79-s + 11/9·81-s − 0.658·83-s − 0.857·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.389117608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.389117608\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 14 T + 90 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_4$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 114 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_4$ | \( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 130 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 24 T + 318 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.702359391795760127122012866109, −8.679063242166751548582150223117, −8.012388052277359715708156319939, −7.77742646207257115461543100573, −7.50579430311463550537508283164, −7.15625015210269498451043918657, −6.80624259319343797894104553597, −6.18323103105265730077311117472, −5.74872083309295907061883205573, −5.44568036984658284447605357533, −4.75394350496884761235543642188, −4.70180550461449963703163615525, −4.30842857922205874238337657125, −3.62421041640382262405841866570, −2.96475698574209663957572115185, −2.95528857473292294870008251077, −2.44203837666168113250173195072, −1.96531733686459999046313640741, −1.12891604678039465084558040210, −0.75809806601510762743361260381,
0.75809806601510762743361260381, 1.12891604678039465084558040210, 1.96531733686459999046313640741, 2.44203837666168113250173195072, 2.95528857473292294870008251077, 2.96475698574209663957572115185, 3.62421041640382262405841866570, 4.30842857922205874238337657125, 4.70180550461449963703163615525, 4.75394350496884761235543642188, 5.44568036984658284447605357533, 5.74872083309295907061883205573, 6.18323103105265730077311117472, 6.80624259319343797894104553597, 7.15625015210269498451043918657, 7.50579430311463550537508283164, 7.77742646207257115461543100573, 8.012388052277359715708156319939, 8.679063242166751548582150223117, 8.702359391795760127122012866109
