| L(s) = 1 | − 9·4-s − 44·7-s + 26·13-s + 17·16-s − 252·19-s − 138·25-s + 396·28-s − 364·31-s − 172·37-s + 192·43-s + 766·49-s − 234·52-s + 1.14e3·61-s + 423·64-s − 1.06e3·67-s − 308·73-s + 2.26e3·76-s − 920·79-s − 1.14e3·91-s + 140·97-s + 1.24e3·100-s − 2.85e3·103-s − 676·109-s − 748·112-s − 2.63e3·121-s + 3.27e3·124-s + 127-s + ⋯ |
| L(s) = 1 | − 9/8·4-s − 2.37·7-s + 0.554·13-s + 0.265·16-s − 3.04·19-s − 1.10·25-s + 2.67·28-s − 2.10·31-s − 0.764·37-s + 0.680·43-s + 2.23·49-s − 0.624·52-s + 2.40·61-s + 0.826·64-s − 1.93·67-s − 0.493·73-s + 3.42·76-s − 1.31·79-s − 1.31·91-s + 0.146·97-s + 1.24·100-s − 2.73·103-s − 0.594·109-s − 0.631·112-s − 1.97·121-s + 2.37·124-s + 0.000698·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 9 T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 138 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 22 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 2634 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3726 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 126 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 23326 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 45978 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 182 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 86 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 59726 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 96 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 74338 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 261466 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 65770 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 574 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 530 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 60370 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 154 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 460 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 1039386 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 661614 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 70 T + p^{3} T^{2} )^{2} \) |
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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
−12.99150470352574584035942427316, −12.57561237509761188203366248690, −11.98073174854902702505941289239, −11.06948416915141318041468229486, −10.54087326293509430015122423146, −10.15143470278430850365840553124, −9.319254954172655942930698156517, −9.304977165246293260978478938253, −8.611307942305103291324962083543, −8.134922373224238986393423304919, −6.98086670748887724328104797280, −6.69620929653497116695108899445, −5.97055871662157869782917407107, −5.52600227951355270372844370243, −4.17973981511170068750084622342, −4.04012168483176216528019367997, −3.20433354821540860461744264412, −2.08744078499639421461408287849, 0, 0,
2.08744078499639421461408287849, 3.20433354821540860461744264412, 4.04012168483176216528019367997, 4.17973981511170068750084622342, 5.52600227951355270372844370243, 5.97055871662157869782917407107, 6.69620929653497116695108899445, 6.98086670748887724328104797280, 8.134922373224238986393423304919, 8.611307942305103291324962083543, 9.304977165246293260978478938253, 9.319254954172655942930698156517, 10.15143470278430850365840553124, 10.54087326293509430015122423146, 11.06948416915141318041468229486, 11.98073174854902702505941289239, 12.57561237509761188203366248690, 12.99150470352574584035942427316
