L(s) = 1 | − 4·11-s − 8·19-s + 4·29-s + 16·31-s + 16·41-s + 10·49-s − 12·59-s + 4·61-s − 24·71-s − 16·79-s − 24·89-s − 20·101-s − 20·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + ⋯ |
L(s) = 1 | − 1.20·11-s − 1.83·19-s + 0.742·29-s + 2.87·31-s + 2.49·41-s + 10/7·49-s − 1.56·59-s + 0.512·61-s − 2.84·71-s − 1.80·79-s − 2.54·89-s − 1.99·101-s − 1.91·109-s − 0.909·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5974690829\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5974690829\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.263482519997862581872838225219, −7.72774562402193866270981461651, −7.60156520541526534678197450761, −6.86359202801641736529325105487, −6.80210727347003648801095748543, −6.38024546511679715111319236614, −5.95702638777339623935685508467, −5.62423476445626674947342543085, −5.48792836538713782322122496049, −4.64753055910622733873644010225, −4.57188879825516906831230876108, −4.17684028632770771531159683527, −4.05875094117863229498805118068, −3.12777500175668417183418102488, −2.81002305177037998925185578590, −2.43824364946363017935671819789, −2.41817320234858887102898167000, −1.27450346935808381323749123661, −1.22641648879376721981641133002, −0.19504476731858127944305836350,
0.19504476731858127944305836350, 1.22641648879376721981641133002, 1.27450346935808381323749123661, 2.41817320234858887102898167000, 2.43824364946363017935671819789, 2.81002305177037998925185578590, 3.12777500175668417183418102488, 4.05875094117863229498805118068, 4.17684028632770771531159683527, 4.57188879825516906831230876108, 4.64753055910622733873644010225, 5.48792836538713782322122496049, 5.62423476445626674947342543085, 5.95702638777339623935685508467, 6.38024546511679715111319236614, 6.80210727347003648801095748543, 6.86359202801641736529325105487, 7.60156520541526534678197450761, 7.72774562402193866270981461651, 8.263482519997862581872838225219