| L(s) = 1 | + 12·11-s + 8·19-s + 12·29-s + 16·31-s − 24·41-s − 49-s − 20·61-s − 12·71-s + 8·79-s + 24·89-s + 24·101-s − 28·109-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 3.61·11-s + 1.83·19-s + 2.22·29-s + 2.87·31-s − 3.74·41-s − 1/7·49-s − 2.56·61-s − 1.42·71-s + 0.900·79-s + 2.54·89-s + 2.38·101-s − 2.68·109-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.756986846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.756986846\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 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
−8.248811662233940481852217313048, −7.990084037998074328060509500241, −7.38827832652121158513470294345, −7.09202293056876309228527824723, −6.58274237140609938530377426043, −6.52017477764552950444850539455, −6.27549974579458084361128064107, −6.01516525757656425775262953203, −5.12358101024344324129832713009, −5.03256300300425476626843118864, −4.43040979552528500862834811718, −4.42735300891593009444764014884, −3.69508927659995564259377443052, −3.48434347903952138590673767717, −2.99850710010784147513581177145, −2.81514690297851522087089700088, −1.66048934383948113939919306771, −1.65603105868881597132470199974, −1.03261491511701919146318739714, −0.74438357632839803568015666113,
0.74438357632839803568015666113, 1.03261491511701919146318739714, 1.65603105868881597132470199974, 1.66048934383948113939919306771, 2.81514690297851522087089700088, 2.99850710010784147513581177145, 3.48434347903952138590673767717, 3.69508927659995564259377443052, 4.42735300891593009444764014884, 4.43040979552528500862834811718, 5.03256300300425476626843118864, 5.12358101024344324129832713009, 6.01516525757656425775262953203, 6.27549974579458084361128064107, 6.52017477764552950444850539455, 6.58274237140609938530377426043, 7.09202293056876309228527824723, 7.38827832652121158513470294345, 7.990084037998074328060509500241, 8.248811662233940481852217313048
