| L(s) = 1 | − 2·5-s + 2·7-s − 2·13-s − 2·17-s − 8·19-s + 10·23-s − 25-s − 4·35-s − 10·37-s − 12·41-s − 6·43-s + 14·47-s + 2·49-s + 2·53-s + 8·59-s + 4·61-s + 4·65-s + 14·67-s + 18·73-s + 16·79-s − 10·83-s + 4·85-s − 4·91-s + 16·95-s − 6·97-s − 12·101-s + 6·103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s − 0.554·13-s − 0.485·17-s − 1.83·19-s + 2.08·23-s − 1/5·25-s − 0.676·35-s − 1.64·37-s − 1.87·41-s − 0.914·43-s + 2.04·47-s + 2/7·49-s + 0.274·53-s + 1.04·59-s + 0.512·61-s + 0.496·65-s + 1.71·67-s + 2.10·73-s + 1.80·79-s − 1.09·83-s + 0.433·85-s − 0.419·91-s + 1.64·95-s − 0.609·97-s − 1.19·101-s + 0.591·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.204171010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.204171010\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−9.675469999657216636874090023670, −9.265891608113872170054977934483, −8.712572433673920965502637146025, −8.451930663381957291601269424263, −8.364233178785128344398035808985, −7.74275847081371917928485496632, −7.23584030757861214270040522074, −6.91581532993709999893223995020, −6.68047208665476241283162642064, −6.16255170325804459690583812212, −5.22115381496844562546035815828, −5.16645523419492230492081799261, −4.86012359757556361137111673558, −3.99630645893780388266443603779, −3.92716459948879044681183296149, −3.33949032735537638685494461587, −2.47738690578755880020146647080, −2.20359469163342630742261339803, −1.40983686235467803294787518095, −0.45067139771728568620706867685,
0.45067139771728568620706867685, 1.40983686235467803294787518095, 2.20359469163342630742261339803, 2.47738690578755880020146647080, 3.33949032735537638685494461587, 3.92716459948879044681183296149, 3.99630645893780388266443603779, 4.86012359757556361137111673558, 5.16645523419492230492081799261, 5.22115381496844562546035815828, 6.16255170325804459690583812212, 6.68047208665476241283162642064, 6.91581532993709999893223995020, 7.23584030757861214270040522074, 7.74275847081371917928485496632, 8.364233178785128344398035808985, 8.451930663381957291601269424263, 8.712572433673920965502637146025, 9.265891608113872170054977934483, 9.675469999657216636874090023670
