| L(s) = 1 | − 9-s + 6·11-s + 25-s + 4·29-s − 6·79-s + 81-s − 6·99-s + 2·109-s + 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 9-s + 6·11-s + 25-s + 4·29-s − 6·79-s + 81-s − 6·99-s + 2·109-s + 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.731686509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.731686509\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.29873762451492276310206779757, −5.84493380529083506430228611510, −5.82269870666592923038987107978, −5.82210702507639454374612060835, −5.65676714092821376924725208124, −5.02876880062995945555256154068, −4.86385865335616126534606945838, −4.61663693854520706083596093571, −4.48208332420018558220197383643, −4.45816290496048886686061733694, −4.27655559800699244905999742179, −3.88288851643952576735041598032, −3.69727127327979788426684530546, −3.54086961742850294143380672087, −3.49810450550787718276406414933, −3.00089807536480077262521595967, −2.93238948403121787022881030754, −2.63846362083070419968824588903, −2.49155783054687746995112091030, −1.91348192347442417615562820089, −1.73596519740185243795890036969, −1.33472995810036280332713341613, −1.28252119891128638503113506162, −0.995031354617777728326785980239, −0.854098872692307461386070715311,
0.854098872692307461386070715311, 0.995031354617777728326785980239, 1.28252119891128638503113506162, 1.33472995810036280332713341613, 1.73596519740185243795890036969, 1.91348192347442417615562820089, 2.49155783054687746995112091030, 2.63846362083070419968824588903, 2.93238948403121787022881030754, 3.00089807536480077262521595967, 3.49810450550787718276406414933, 3.54086961742850294143380672087, 3.69727127327979788426684530546, 3.88288851643952576735041598032, 4.27655559800699244905999742179, 4.45816290496048886686061733694, 4.48208332420018558220197383643, 4.61663693854520706083596093571, 4.86385865335616126534606945838, 5.02876880062995945555256154068, 5.65676714092821376924725208124, 5.82210702507639454374612060835, 5.82269870666592923038987107978, 5.84493380529083506430228611510, 6.29873762451492276310206779757
