| L(s) = 1 | − 4·3-s − 3·5-s − 6·7-s + 7·9-s − 13-s + 12·15-s − 9·17-s − 2·19-s + 24·21-s − 6·23-s + 25-s − 4·27-s + 29-s + 2·31-s + 18·35-s + 2·37-s + 4·39-s − 8·43-s − 21·45-s + 15·49-s + 36·51-s + 7·53-s + 8·57-s − 2·59-s + 3·61-s − 42·63-s + 3·65-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1.34·5-s − 2.26·7-s + 7/3·9-s − 0.277·13-s + 3.09·15-s − 2.18·17-s − 0.458·19-s + 5.23·21-s − 1.25·23-s + 1/5·25-s − 0.769·27-s + 0.185·29-s + 0.359·31-s + 3.04·35-s + 0.328·37-s + 0.640·39-s − 1.21·43-s − 3.13·45-s + 15/7·49-s + 5.04·51-s + 0.961·53-s + 1.05·57-s − 0.260·59-s + 0.384·61-s − 5.29·63-s + 0.372·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 193 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 17 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 6 T + 3 p T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 9 T + 46 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T - 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 11 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 29 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3 T + 66 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T - 53 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 56 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 21 T + 228 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T - 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−16.1483051938, −15.7835329652, −15.2833487544, −14.9698930036, −13.8767263490, −13.3649405453, −12.9328332489, −12.5317509922, −11.9910524239, −11.7091093966, −11.3873667344, −10.7413888829, −10.3721672931, −9.84563669314, −9.23619427149, −8.55134998213, −7.86833193035, −6.89477780439, −6.73115701283, −6.24593319538, −5.81129692840, −4.95432871817, −4.23397773651, −3.76231070724, −2.63730795332, 0, 0,
2.63730795332, 3.76231070724, 4.23397773651, 4.95432871817, 5.81129692840, 6.24593319538, 6.73115701283, 6.89477780439, 7.86833193035, 8.55134998213, 9.23619427149, 9.84563669314, 10.3721672931, 10.7413888829, 11.3873667344, 11.7091093966, 11.9910524239, 12.5317509922, 12.9328332489, 13.3649405453, 13.8767263490, 14.9698930036, 15.2833487544, 15.7835329652, 16.1483051938
