| L(s) = 1 | − 2-s + 4-s − 8-s + 6·11-s − 5·13-s + 16-s − 6·22-s + 3·23-s + 8·25-s + 5·26-s − 32-s + 37-s + 6·44-s − 3·46-s + 12·47-s − 10·49-s − 8·50-s − 5·52-s + 3·59-s − 11·61-s + 64-s − 3·71-s + 4·73-s − 74-s + 12·83-s − 6·88-s + 3·92-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1.80·11-s − 1.38·13-s + 1/4·16-s − 1.27·22-s + 0.625·23-s + 8/5·25-s + 0.980·26-s − 0.176·32-s + 0.164·37-s + 0.904·44-s − 0.442·46-s + 1.75·47-s − 1.42·49-s − 1.13·50-s − 0.693·52-s + 0.390·59-s − 1.40·61-s + 1/8·64-s − 0.356·71-s + 0.468·73-s − 0.116·74-s + 1.31·83-s − 0.639·88-s + 0.312·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.298639604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.298639604\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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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
−9.094465745776204016944600718579, −8.760428423359313509815755359349, −8.193402536150836816173769499853, −7.55715890902350916938496160193, −7.15086592553495636081374704142, −6.80034227494803708852389773913, −6.31658006260421554962433215703, −5.78391809478288755262203915963, −4.93729801362432174728113527852, −4.65523994208989521754605857959, −3.88569328110345092532109758989, −3.19425764883904579339908438440, −2.57411427643257382919325899440, −1.70798197754385916288770338481, −0.846795853288317835391085768782,
0.846795853288317835391085768782, 1.70798197754385916288770338481, 2.57411427643257382919325899440, 3.19425764883904579339908438440, 3.88569328110345092532109758989, 4.65523994208989521754605857959, 4.93729801362432174728113527852, 5.78391809478288755262203915963, 6.31658006260421554962433215703, 6.80034227494803708852389773913, 7.15086592553495636081374704142, 7.55715890902350916938496160193, 8.193402536150836816173769499853, 8.760428423359313509815755359349, 9.094465745776204016944600718579
