| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 4·6-s − 8·7-s − 4·8-s + 9-s − 6·12-s + 16·14-s + 5·16-s − 2·18-s + 2·19-s + 16·21-s + 8·24-s + 8·25-s + 4·27-s − 24·28-s + 12·29-s − 6·32-s + 3·36-s − 4·38-s − 32·42-s + 4·43-s − 10·48-s + 34·49-s − 16·50-s − 12·53-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 3.02·7-s − 1.41·8-s + 1/3·9-s − 1.73·12-s + 4.27·14-s + 5/4·16-s − 0.471·18-s + 0.458·19-s + 3.49·21-s + 1.63·24-s + 8/5·25-s + 0.769·27-s − 4.53·28-s + 2.22·29-s − 1.06·32-s + 1/2·36-s − 0.648·38-s − 4.93·42-s + 0.609·43-s − 1.44·48-s + 34/7·49-s − 2.26·50-s − 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2090591793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2090591793\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 19 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
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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
−14.17461496914506874884294786783, −12.86554969930770695368190381367, −12.59996735577287581826995145004, −12.39727882625190788016740595340, −11.72666334402644106659205396212, −11.01841174469738338022426905919, −10.56265684308273267318571520182, −10.13004918113045919660344999085, −9.687257480288723321972873330012, −9.206044813673965073257274287356, −8.732450706857503879431768376690, −7.88739778826632931859306112493, −6.99016355562247833280236354883, −6.56900575356071618745082935569, −6.36644239338775995376605019153, −5.72842745695832435114139827872, −4.67072515898436690904446070940, −3.13805947770897882212007565114, −2.97323078393205506761813742798, −0.66816671282063161139195356884,
0.66816671282063161139195356884, 2.97323078393205506761813742798, 3.13805947770897882212007565114, 4.67072515898436690904446070940, 5.72842745695832435114139827872, 6.36644239338775995376605019153, 6.56900575356071618745082935569, 6.99016355562247833280236354883, 7.88739778826632931859306112493, 8.732450706857503879431768376690, 9.206044813673965073257274287356, 9.687257480288723321972873330012, 10.13004918113045919660344999085, 10.56265684308273267318571520182, 11.01841174469738338022426905919, 11.72666334402644106659205396212, 12.39727882625190788016740595340, 12.59996735577287581826995145004, 12.86554969930770695368190381367, 14.17461496914506874884294786783
