| L(s) = 1 | + 2·5-s − 4·7-s − 4·11-s + 2·13-s + 12·17-s + 8·19-s + 5·25-s + 2·29-s + 4·31-s − 8·35-s − 4·37-s + 2·41-s + 4·43-s − 8·47-s + 7·49-s − 20·53-s − 8·55-s + 4·59-s − 6·61-s + 4·65-s + 4·67-s − 32·71-s − 12·73-s + 16·77-s + 4·79-s − 12·83-s + 24·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s − 1.20·11-s + 0.554·13-s + 2.91·17-s + 1.83·19-s + 25-s + 0.371·29-s + 0.718·31-s − 1.35·35-s − 0.657·37-s + 0.312·41-s + 0.609·43-s − 1.16·47-s + 49-s − 2.74·53-s − 1.07·55-s + 0.520·59-s − 0.768·61-s + 0.496·65-s + 0.488·67-s − 3.79·71-s − 1.40·73-s + 1.82·77-s + 0.450·79-s − 1.31·83-s + 2.60·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.439511135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.439511135\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−9.353773918829635163119583115159, −8.523196761884638693677920060893, −8.466770107578634234459457748071, −7.82563842306841597038728593252, −7.55126874611262419808969959611, −7.18481242833963500625169124498, −6.86166440945316610047332621942, −6.10979780039144253905857067493, −6.01006486638016017582975513082, −5.53378037272390217592388190967, −5.51606420923729636508185604890, −4.71883769099416620556295262357, −4.51376439229060954432598363816, −3.49839209218241177546766190160, −3.11594811809659479814376814325, −3.11210321750689923638245189346, −2.75777918642639635223666845750, −1.61243274288161144511544500488, −1.33153106367843797436547517728, −0.54623318411897654023694048051,
0.54623318411897654023694048051, 1.33153106367843797436547517728, 1.61243274288161144511544500488, 2.75777918642639635223666845750, 3.11210321750689923638245189346, 3.11594811809659479814376814325, 3.49839209218241177546766190160, 4.51376439229060954432598363816, 4.71883769099416620556295262357, 5.51606420923729636508185604890, 5.53378037272390217592388190967, 6.01006486638016017582975513082, 6.10979780039144253905857067493, 6.86166440945316610047332621942, 7.18481242833963500625169124498, 7.55126874611262419808969959611, 7.82563842306841597038728593252, 8.466770107578634234459457748071, 8.523196761884638693677920060893, 9.353773918829635163119583115159
