| L(s) = 1 | + 7·4-s + 52·13-s − 15·16-s + 90·17-s − 324·23-s + 169·25-s + 288·29-s − 194·43-s + 461·49-s + 364·52-s + 828·53-s + 752·61-s − 553·64-s + 630·68-s − 1.66e3·79-s − 2.26e3·92-s + 1.18e3·100-s − 792·101-s + 364·103-s + 1.22e3·107-s − 180·113-s + 2.01e3·116-s + 358·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 7/8·4-s + 1.10·13-s − 0.234·16-s + 1.28·17-s − 2.93·23-s + 1.35·25-s + 1.84·29-s − 0.688·43-s + 1.34·49-s + 0.970·52-s + 2.14·53-s + 1.57·61-s − 1.08·64-s + 1.12·68-s − 2.36·79-s − 2.57·92-s + 1.18·100-s − 0.780·101-s + 0.348·103-s + 1.10·107-s − 0.149·113-s + 1.61·116-s + 0.268·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-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
| \(L(2)\) |
\(\approx\) |
\(2.782504586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.782504586\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - 4 p T + p^{3} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 7 T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 169 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 461 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 358 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 45 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 13682 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 162 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 144 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 10114 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 9497 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 100978 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 97 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 195325 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 414 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 138274 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 376 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 600230 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 588373 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 592 T + p^{3} T^{2} )( 1 + 592 T + p^{3} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 830 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 951730 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1218094 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 1099442 T^{2} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.60398853301899354166869039618, −12.52770032991022459607411272654, −12.28523262816560121608618679314, −11.68133189128651701899748085259, −11.44242563757729249643944176193, −10.58563031057546712771130785613, −10.16531555265337622353998003196, −9.963633542248425180674785622269, −8.832199846294356124752250965549, −8.455853132209318819267159524074, −7.960186134564257476210832150253, −7.13523943846753024282146751179, −6.71160424612649585510244407896, −5.94530028008584231044139187000, −5.62794722185864868354909014395, −4.48966436094833921158187186980, −3.80145045542041564237822807466, −2.91609446902805625928921635242, −2.04411486935465217774559345651, −0.942480408471929766551411869705,
0.942480408471929766551411869705, 2.04411486935465217774559345651, 2.91609446902805625928921635242, 3.80145045542041564237822807466, 4.48966436094833921158187186980, 5.62794722185864868354909014395, 5.94530028008584231044139187000, 6.71160424612649585510244407896, 7.13523943846753024282146751179, 7.960186134564257476210832150253, 8.455853132209318819267159524074, 8.832199846294356124752250965549, 9.963633542248425180674785622269, 10.16531555265337622353998003196, 10.58563031057546712771130785613, 11.44242563757729249643944176193, 11.68133189128651701899748085259, 12.28523262816560121608618679314, 12.52770032991022459607411272654, 13.60398853301899354166869039618
