| L(s) = 1 | + 6·9-s + 8·11-s − 10·25-s − 24·43-s − 7·49-s − 8·67-s + 27·81-s + 48·99-s + 40·107-s − 4·113-s + 26·121-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 + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 2·9-s + 2.41·11-s − 2·25-s − 3.65·43-s − 49-s − 0.977·67-s + 3·81-s + 4.82·99-s + 3.86·107-s − 0.376·113-s + 2.36·121-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 + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.748146366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.748146366\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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
−12.32180775967608868365646478578, −12.06557483860527034299620531922, −11.50064929263654063357137787404, −11.34436108247883167850609453958, −10.31051710828699044062156525461, −10.04871488055362295682351441334, −9.582358494613024752970500930116, −9.312480829385573256702764495218, −8.546395830076278245343632819159, −8.083173450993270659605003738941, −7.16150935108467611770633917749, −7.15168892393948979255100184924, −6.23831579269273872917902894561, −6.21582404203390083723612488505, −4.92680883399711132158191577086, −4.53270465427190292181413969951, −3.67547683248527082149287337927, −3.62797910414392303563635931634, −1.83878120835424374972243439361, −1.45605785642116846050891671960,
1.45605785642116846050891671960, 1.83878120835424374972243439361, 3.62797910414392303563635931634, 3.67547683248527082149287337927, 4.53270465427190292181413969951, 4.92680883399711132158191577086, 6.21582404203390083723612488505, 6.23831579269273872917902894561, 7.15168892393948979255100184924, 7.16150935108467611770633917749, 8.083173450993270659605003738941, 8.546395830076278245343632819159, 9.312480829385573256702764495218, 9.582358494613024752970500930116, 10.04871488055362295682351441334, 10.31051710828699044062156525461, 11.34436108247883167850609453958, 11.50064929263654063357137787404, 12.06557483860527034299620531922, 12.32180775967608868365646478578
