| L(s) = 1 | + 5-s − 2·7-s − 3·11-s + 4·13-s + 12·17-s − 14·19-s − 6·23-s − 3·29-s − 5·31-s − 2·35-s − 8·37-s − 3·41-s − 8·43-s + 7·49-s − 12·53-s − 3·55-s + 3·59-s − 14·61-s + 4·65-s − 2·67-s + 30·71-s − 20·73-s + 6·77-s − 8·79-s + 12·85-s + 30·89-s − 8·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 0.904·11-s + 1.10·13-s + 2.91·17-s − 3.21·19-s − 1.25·23-s − 0.557·29-s − 0.898·31-s − 0.338·35-s − 1.31·37-s − 0.468·41-s − 1.21·43-s + 49-s − 1.64·53-s − 0.404·55-s + 0.390·59-s − 1.79·61-s + 0.496·65-s − 0.244·67-s + 3.56·71-s − 2.34·73-s + 0.683·77-s − 0.900·79-s + 1.30·85-s + 3.17·89-s − 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.042306221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.042306221\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.733501807364841297367726675989, −9.032121123292942369102759978647, −8.935928026128662971323824572693, −8.198718469908847260474219690547, −8.182246885703072703659613329577, −7.70583761731472020500562330291, −7.24709866484030318812929750398, −6.56027937471453598538172397904, −6.38769295441514331692446752554, −5.82200167744826480198156111725, −5.77592695311896916236220881413, −5.11486372111443652050799840450, −4.70939468512892644145029918618, −3.78653693355846040566700122672, −3.78295698920017769567034993412, −3.23651932799727775082472775663, −2.63473038238521854758496839042, −1.76612952536814350740930298696, −1.70565649923453857635540106232, −0.37656550841158718199288990281,
0.37656550841158718199288990281, 1.70565649923453857635540106232, 1.76612952536814350740930298696, 2.63473038238521854758496839042, 3.23651932799727775082472775663, 3.78295698920017769567034993412, 3.78653693355846040566700122672, 4.70939468512892644145029918618, 5.11486372111443652050799840450, 5.77592695311896916236220881413, 5.82200167744826480198156111725, 6.38769295441514331692446752554, 6.56027937471453598538172397904, 7.24709866484030318812929750398, 7.70583761731472020500562330291, 8.182246885703072703659613329577, 8.198718469908847260474219690547, 8.935928026128662971323824572693, 9.032121123292942369102759978647, 9.733501807364841297367726675989
