| L(s) = 1 | + 2·7-s + 2·9-s + 12·17-s − 8·23-s + 6·25-s − 16·31-s − 12·41-s + 3·49-s + 4·63-s − 32·71-s + 12·73-s + 32·79-s − 5·81-s + 28·89-s − 4·97-s + 32·103-s − 20·113-s + 24·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 2/3·9-s + 2.91·17-s − 1.66·23-s + 6/5·25-s − 2.87·31-s − 1.87·41-s + 3/7·49-s + 0.503·63-s − 3.79·71-s + 1.40·73-s + 3.60·79-s − 5/9·81-s + 2.96·89-s − 0.406·97-s + 3.15·103-s − 1.88·113-s + 2.20·119-s + 2·121-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 + 1.94·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 802816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 802816 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.489163062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.489163062\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−10.40127408081591978929773660381, −9.804155830162207405517004001813, −9.719104836408782915988558686051, −8.956202068870515196987246586886, −8.770579645144117436471212270114, −7.982361374988612303386556882570, −7.86427692448794406308763970929, −7.42476312026384558923739073067, −7.11890937961443462564381487319, −6.42032517500321750045528698838, −5.92045022256968320597341900784, −5.42625603075705964719964372265, −5.15865914961395090343056253630, −4.62507158999568088575334581251, −3.91536554176197935018058155674, −3.44151730792668139078293444251, −3.17008152437631666866722841487, −1.88324751361067601798360210873, −1.77870064030860721788297359816, −0.797204427107746605432915597720,
0.797204427107746605432915597720, 1.77870064030860721788297359816, 1.88324751361067601798360210873, 3.17008152437631666866722841487, 3.44151730792668139078293444251, 3.91536554176197935018058155674, 4.62507158999568088575334581251, 5.15865914961395090343056253630, 5.42625603075705964719964372265, 5.92045022256968320597341900784, 6.42032517500321750045528698838, 7.11890937961443462564381487319, 7.42476312026384558923739073067, 7.86427692448794406308763970929, 7.982361374988612303386556882570, 8.770579645144117436471212270114, 8.956202068870515196987246586886, 9.719104836408782915988558686051, 9.804155830162207405517004001813, 10.40127408081591978929773660381
