L(s) = 1 | + 2·2-s − 4-s + 2·5-s − 8·8-s + 2·9-s + 4·10-s − 6·13-s − 7·16-s + 4·18-s − 2·20-s − 25-s − 12·26-s + 12·29-s + 14·32-s − 2·36-s − 12·37-s − 16·40-s + 4·45-s + 16·47-s − 14·49-s − 2·50-s + 6·52-s + 24·58-s + 12·61-s + 35·64-s − 12·65-s − 24·67-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 1/2·4-s + 0.894·5-s − 2.82·8-s + 2/3·9-s + 1.26·10-s − 1.66·13-s − 7/4·16-s + 0.942·18-s − 0.447·20-s − 1/5·25-s − 2.35·26-s + 2.22·29-s + 2.47·32-s − 1/3·36-s − 1.97·37-s − 2.52·40-s + 0.596·45-s + 2.33·47-s − 2·49-s − 0.282·50-s + 0.832·52-s + 3.15·58-s + 1.53·61-s + 35/8·64-s − 1.48·65-s − 2.93·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.221798417\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.221798417\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−14.93053022598393681489846106157, −14.34862516746341784522932702950, −13.92357669379673350762503868502, −13.78823751106707503594314175371, −12.93251967907106945395356216934, −12.71714119571974731062634462663, −11.98239077020634614152784228637, −11.90311223456943156683204875337, −10.33586677104772337841262224795, −10.15382394013077472054980476610, −9.463934432947145288101362720902, −8.972634087180848766990541926844, −8.260808297600080955067953336526, −7.23591734839184107054042839302, −6.42372356969681330659811031444, −5.73514302994749178233236282911, −4.88156617889206626562591184346, −4.72173132163098801836255991991, −3.64189765661446866968181750927, −2.58431702142523155261267406595,
2.58431702142523155261267406595, 3.64189765661446866968181750927, 4.72173132163098801836255991991, 4.88156617889206626562591184346, 5.73514302994749178233236282911, 6.42372356969681330659811031444, 7.23591734839184107054042839302, 8.260808297600080955067953336526, 8.972634087180848766990541926844, 9.463934432947145288101362720902, 10.15382394013077472054980476610, 10.33586677104772337841262224795, 11.90311223456943156683204875337, 11.98239077020634614152784228637, 12.71714119571974731062634462663, 12.93251967907106945395356216934, 13.78823751106707503594314175371, 13.92357669379673350762503868502, 14.34862516746341784522932702950, 14.93053022598393681489846106157