| L(s) = 1 | + 2·5-s + 2·9-s − 8·11-s − 8·19-s − 25-s − 4·29-s − 4·41-s + 4·45-s − 16·55-s − 24·59-s + 20·61-s + 16·71-s − 32·79-s − 5·81-s + 12·89-s − 16·95-s − 16·99-s − 12·101-s + 12·109-s + 26·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 2/3·9-s − 2.41·11-s − 1.83·19-s − 1/5·25-s − 0.742·29-s − 0.624·41-s + 0.596·45-s − 2.15·55-s − 3.12·59-s + 2.56·61-s + 1.89·71-s − 3.60·79-s − 5/9·81-s + 1.27·89-s − 1.64·95-s − 1.60·99-s − 1.19·101-s + 1.14·109-s + 2.36·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9842714143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9842714143\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−9.637244616815914765340100375155, −8.848426049592365618984359381360, −8.593897471046692907627939762493, −8.235585501317652893810304862043, −7.85390848862261974976355825286, −7.32814370447557757601087272345, −7.20294575888161820302129955739, −6.53884280212563627203727225074, −6.11495671370011548301406368150, −5.86056772947807781412996216914, −5.29798925345426560110549974082, −5.02469437549328439557199908410, −4.59537567361456782616078085091, −4.05010593479822587704589812835, −3.58485400635582933248857258888, −2.75555881058552054109776838878, −2.57097691746894201931413493371, −1.93849356072506714350232402441, −1.58107359650683069944637109463, −0.33352152405547286101464605490,
0.33352152405547286101464605490, 1.58107359650683069944637109463, 1.93849356072506714350232402441, 2.57097691746894201931413493371, 2.75555881058552054109776838878, 3.58485400635582933248857258888, 4.05010593479822587704589812835, 4.59537567361456782616078085091, 5.02469437549328439557199908410, 5.29798925345426560110549974082, 5.86056772947807781412996216914, 6.11495671370011548301406368150, 6.53884280212563627203727225074, 7.20294575888161820302129955739, 7.32814370447557757601087272345, 7.85390848862261974976355825286, 8.235585501317652893810304862043, 8.593897471046692907627939762493, 8.848426049592365618984359381360, 9.637244616815914765340100375155
