| L(s) = 1 | + 2·5-s − 2·13-s + 6·17-s − 25-s − 8·29-s − 14·37-s + 16·41-s − 18·53-s − 4·65-s + 22·73-s − 9·81-s + 12·85-s + 26·97-s + 12·109-s − 2·113-s − 22·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.554·13-s + 1.45·17-s − 1/5·25-s − 1.48·29-s − 2.30·37-s + 2.49·41-s − 2.47·53-s − 0.496·65-s + 2.57·73-s − 81-s + 1.30·85-s + 2.63·97-s + 1.14·109-s − 0.188·113-s − 2·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.298856930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.298856930\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
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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
−10.05879794562371001412071309452, −9.382472361501381457618097390178, −9.219060541222198231611306318034, −8.863510508636534688063042677467, −8.062749379156499831713267310079, −7.83146047766568019645839035004, −7.55804970294160445959981322978, −6.97380082819873646303314417643, −6.58776436759169355051601709330, −5.98158107861579876239433546278, −5.74677689663861871884237201582, −5.27988271403578315098341106402, −4.95350834815705090788492125697, −4.30204232496745581832036766413, −3.68735195784165874322770106795, −3.29380686358137115966435644102, −2.71012991301328914697046270844, −1.92679274233205558666457369245, −1.68741738297018311505063943572, −0.63131316802742449880273092907,
0.63131316802742449880273092907, 1.68741738297018311505063943572, 1.92679274233205558666457369245, 2.71012991301328914697046270844, 3.29380686358137115966435644102, 3.68735195784165874322770106795, 4.30204232496745581832036766413, 4.95350834815705090788492125697, 5.27988271403578315098341106402, 5.74677689663861871884237201582, 5.98158107861579876239433546278, 6.58776436759169355051601709330, 6.97380082819873646303314417643, 7.55804970294160445959981322978, 7.83146047766568019645839035004, 8.062749379156499831713267310079, 8.863510508636534688063042677467, 9.219060541222198231611306318034, 9.382472361501381457618097390178, 10.05879794562371001412071309452
