L(s) = 1 | + 7-s + 4·9-s + 2·11-s + 6·23-s − 2·25-s − 2·29-s + 14·37-s + 10·43-s + 49-s + 4·53-s + 4·63-s + 4·67-s + 12·71-s + 2·77-s + 7·81-s + 8·99-s + 12·107-s + 6·109-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 6·161-s + ⋯ |
L(s) = 1 | + 0.377·7-s + 4/3·9-s + 0.603·11-s + 1.25·23-s − 2/5·25-s − 0.371·29-s + 2.30·37-s + 1.52·43-s + 1/7·49-s + 0.549·53-s + 0.503·63-s + 0.488·67-s + 1.42·71-s + 0.227·77-s + 7/9·81-s + 0.804·99-s + 1.16·107-s + 0.574·109-s − 1.63·121-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 + 0.472·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.120879684\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.120879684\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
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$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
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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
−7.85682464165540802403552444351, −7.43092074254597553488682277284, −7.21987900761935729020715214367, −6.62991578485059005464543438416, −6.31036940487758145552263989909, −5.75649159255783601210481235838, −5.29748585758289986421126568532, −4.64599244228357963317943642414, −4.43327569551569944762491846130, −3.87251519430086561836938306178, −3.48873456518669671206626509996, −2.59443888659994177654662127781, −2.20501522187753158347170165946, −1.30164515059168820566347798712, −0.898548702185509465307047713875,
0.898548702185509465307047713875, 1.30164515059168820566347798712, 2.20501522187753158347170165946, 2.59443888659994177654662127781, 3.48873456518669671206626509996, 3.87251519430086561836938306178, 4.43327569551569944762491846130, 4.64599244228357963317943642414, 5.29748585758289986421126568532, 5.75649159255783601210481235838, 6.31036940487758145552263989909, 6.62991578485059005464543438416, 7.21987900761935729020715214367, 7.43092074254597553488682277284, 7.85682464165540802403552444351