| L(s) = 1 | + 2·5-s − 7-s + 2·11-s − 13-s − 12·17-s − 10·19-s − 6·23-s + 5·25-s + 8·29-s − 8·31-s − 2·35-s − 10·37-s + 8·41-s + 4·43-s + 10·47-s + 7·49-s − 8·53-s + 4·55-s − 14·59-s − 3·61-s − 2·65-s + 13·67-s − 8·71-s + 18·73-s − 2·77-s − 11·79-s + 12·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s + 0.603·11-s − 0.277·13-s − 2.91·17-s − 2.29·19-s − 1.25·23-s + 25-s + 1.48·29-s − 1.43·31-s − 0.338·35-s − 1.64·37-s + 1.24·41-s + 0.609·43-s + 1.45·47-s + 49-s − 1.09·53-s + 0.539·55-s − 1.82·59-s − 0.384·61-s − 0.248·65-s + 1.58·67-s − 0.949·71-s + 2.10·73-s − 0.227·77-s − 1.23·79-s + 1.31·83-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
| \(L(1)\) |
\(\approx\) |
\(0.8392769240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8392769240\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 3 T - 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−9.009022716472264418779033159351, −8.649944092255145522756546529039, −8.634537837675451359002650181598, −8.046976466648692633894050132124, −7.44434627837536520937000682171, −6.90399314407430787810710999240, −6.82889127460352438549083489107, −6.33343729755726719893649220422, −6.04797544319859575536319676420, −5.87392053169782770705470233369, −5.05229215526127219463454538780, −4.59009484245182620070375389666, −4.45531722659863800064174109647, −3.89219501947325747785986371395, −3.54820953935455949684083733370, −2.58888551232916792624614062988, −2.20401594086502327641214219011, −2.19215591558151794541522193288, −1.38136835743757724794755140731, −0.28553088501887923982534574652,
0.28553088501887923982534574652, 1.38136835743757724794755140731, 2.19215591558151794541522193288, 2.20401594086502327641214219011, 2.58888551232916792624614062988, 3.54820953935455949684083733370, 3.89219501947325747785986371395, 4.45531722659863800064174109647, 4.59009484245182620070375389666, 5.05229215526127219463454538780, 5.87392053169782770705470233369, 6.04797544319859575536319676420, 6.33343729755726719893649220422, 6.82889127460352438549083489107, 6.90399314407430787810710999240, 7.44434627837536520937000682171, 8.046976466648692633894050132124, 8.634537837675451359002650181598, 8.649944092255145522756546529039, 9.009022716472264418779033159351
