| L(s) = 1 | + 10·11-s − 12·19-s − 18·29-s − 8·31-s + 8·41-s − 49-s − 16·59-s − 16·61-s − 16·71-s − 26·79-s + 8·89-s − 12·101-s + 6·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 17·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 3.01·11-s − 2.75·19-s − 3.34·29-s − 1.43·31-s + 1.24·41-s − 1/7·49-s − 2.08·59-s − 2.04·61-s − 1.89·71-s − 2.92·79-s + 0.847·89-s − 1.19·101-s + 0.574·109-s + 4.81·121-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 + 1.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-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
| \(L(1)\) |
\(\approx\) |
\(0.3226155249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3226155249\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 93 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 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
−8.478417479436180116409165275107, −7.79118981317633536049255396602, −7.34283278627811604596788535434, −7.11972690894318710197543067944, −6.96545245413929252753271003510, −6.24849622248479925469332702328, −6.15795097553311578530832383953, −5.80381895235453923261200190360, −5.71356802344362859528979674599, −4.75178624535792741349731805971, −4.46687571586783342039539752513, −4.10861208738486395686089055173, −4.04066243561118890196872887829, −3.35460838506374001683745741926, −3.30950576150913836515926375944, −2.33848180506657473760362020051, −1.97846649761999451410251225036, −1.44663021611817929530769155332, −1.42888724028672763865249997475, −0.13550983967192206174570100412,
0.13550983967192206174570100412, 1.42888724028672763865249997475, 1.44663021611817929530769155332, 1.97846649761999451410251225036, 2.33848180506657473760362020051, 3.30950576150913836515926375944, 3.35460838506374001683745741926, 4.04066243561118890196872887829, 4.10861208738486395686089055173, 4.46687571586783342039539752513, 4.75178624535792741349731805971, 5.71356802344362859528979674599, 5.80381895235453923261200190360, 6.15795097553311578530832383953, 6.24849622248479925469332702328, 6.96545245413929252753271003510, 7.11972690894318710197543067944, 7.34283278627811604596788535434, 7.79118981317633536049255396602, 8.478417479436180116409165275107
