| L(s) = 1 | − 4·5-s − 9-s + 4·11-s − 16·19-s + 11·25-s + 16·29-s + 4·41-s + 4·45-s + 10·49-s − 16·55-s + 12·59-s − 4·61-s + 8·71-s − 16·79-s + 81-s − 12·89-s + 64·95-s − 4·99-s − 12·109-s − 10·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s − 64·145-s + 149-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1/3·9-s + 1.20·11-s − 3.67·19-s + 11/5·25-s + 2.97·29-s + 0.624·41-s + 0.596·45-s + 10/7·49-s − 2.15·55-s + 1.56·59-s − 0.512·61-s + 0.949·71-s − 1.80·79-s + 1/9·81-s − 1.27·89-s + 6.56·95-s − 0.402·99-s − 1.14·109-s − 0.909·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.31·145-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9303908879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9303908879\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 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
−10.26870671570928651544769329884, −10.08363037580864116350997436415, −9.005182915485349800402921961099, −8.938326430419058969539559281165, −8.524708963394712764702786462144, −8.263121450869900726581895515883, −7.88366258694318355537920330272, −7.25810532710205171025243442797, −6.69776241199813181284086333673, −6.51494980513937177174276862019, −6.25788656631927173474389913661, −5.40303945019765498410426269798, −4.73018368441145937045797029115, −4.29048960202738611119589370171, −4.04529255878406932176264212297, −3.75625188599162295958960767058, −2.66281176407388149517619154979, −2.58222617141465845178748427393, −1.38493726723817759877125434325, −0.47867887922774400161696325136,
0.47867887922774400161696325136, 1.38493726723817759877125434325, 2.58222617141465845178748427393, 2.66281176407388149517619154979, 3.75625188599162295958960767058, 4.04529255878406932176264212297, 4.29048960202738611119589370171, 4.73018368441145937045797029115, 5.40303945019765498410426269798, 6.25788656631927173474389913661, 6.51494980513937177174276862019, 6.69776241199813181284086333673, 7.25810532710205171025243442797, 7.88366258694318355537920330272, 8.263121450869900726581895515883, 8.524708963394712764702786462144, 8.938326430419058969539559281165, 9.005182915485349800402921961099, 10.08363037580864116350997436415, 10.26870671570928651544769329884
