| L(s) = 1 | − 4·9-s − 12·11-s − 8·23-s − 2·25-s − 12·29-s + 4·37-s − 20·43-s − 4·53-s − 8·67-s + 24·71-s + 8·79-s + 7·81-s + 48·99-s − 8·107-s − 20·109-s + 8·113-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + ⋯ |
| L(s) = 1 | − 4/3·9-s − 3.61·11-s − 1.66·23-s − 2/5·25-s − 2.22·29-s + 0.657·37-s − 3.04·43-s − 0.549·53-s − 0.977·67-s + 2.84·71-s + 0.900·79-s + 7/9·81-s + 4.82·99-s − 0.773·107-s − 1.91·109-s + 0.752·113-s + 7.81·121-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 + 6/13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 164 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 160 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
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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
−10.02048325708120582024979437385, −9.845809419307233483499829762170, −9.326714526433421061268572171439, −8.608238699688466466833484896727, −8.204114441276929531445016516329, −7.999777301734174932209907910997, −7.70842157424562848230327746919, −7.26620613159574683228237823772, −6.42822498836810409780446243939, −6.03984150447999346896937959114, −5.39077000624782911690119936583, −5.33360296629171073414900523560, −4.94881687349208534225804832367, −4.07090503218090234482268790212, −3.36435802030019752953020125633, −2.99279108425455303647003011002, −2.23667764518565054470265041151, −2.04423569724364312331338784291, 0, 0,
2.04423569724364312331338784291, 2.23667764518565054470265041151, 2.99279108425455303647003011002, 3.36435802030019752953020125633, 4.07090503218090234482268790212, 4.94881687349208534225804832367, 5.33360296629171073414900523560, 5.39077000624782911690119936583, 6.03984150447999346896937959114, 6.42822498836810409780446243939, 7.26620613159574683228237823772, 7.70842157424562848230327746919, 7.999777301734174932209907910997, 8.204114441276929531445016516329, 8.608238699688466466833484896727, 9.326714526433421061268572171439, 9.845809419307233483499829762170, 10.02048325708120582024979437385
