| L(s) = 1 | + 3·4-s + 6·9-s + 8·11-s + 5·16-s − 4·29-s + 18·36-s + 24·44-s + 3·64-s + 32·71-s − 16·79-s + 27·81-s + 48·99-s − 36·109-s − 12·116-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 30·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 2·9-s + 2.41·11-s + 5/4·16-s − 0.742·29-s + 3·36-s + 3.61·44-s + 3/8·64-s + 3.79·71-s − 1.80·79-s + 3·81-s + 4.82·99-s − 3.44·109-s − 1.11·116-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.232771811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.232771811\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−9.771250188868561795391688243002, −9.724662915621575717143485544974, −9.088080049500239296415354148394, −8.952772387457675256535099877117, −8.079528927219190463262967036019, −7.83316777882651927873725804425, −7.24030895518800721395748600639, −7.00666616410248295909397260561, −6.60973483690631765570905823420, −6.46895163930502220218130958741, −5.97896574798438697490533614779, −5.27964669881712485466281931799, −4.77427583873579081767954459657, −3.99216095553573431735843439729, −3.94471686331882774899431803123, −3.47185485966901585591656140513, −2.55727884186039395969986671853, −2.02495550452506870676685756872, −1.36321925300395650269465102503, −1.20481275212509939452365056019,
1.20481275212509939452365056019, 1.36321925300395650269465102503, 2.02495550452506870676685756872, 2.55727884186039395969986671853, 3.47185485966901585591656140513, 3.94471686331882774899431803123, 3.99216095553573431735843439729, 4.77427583873579081767954459657, 5.27964669881712485466281931799, 5.97896574798438697490533614779, 6.46895163930502220218130958741, 6.60973483690631765570905823420, 7.00666616410248295909397260561, 7.24030895518800721395748600639, 7.83316777882651927873725804425, 8.079528927219190463262967036019, 8.952772387457675256535099877117, 9.088080049500239296415354148394, 9.724662915621575717143485544974, 9.771250188868561795391688243002
