| L(s) = 1 | + 18·5-s + 36·11-s + 20·13-s − 18·17-s − 100·19-s + 72·23-s + 125·25-s + 468·29-s − 16·31-s + 226·37-s + 180·41-s + 904·43-s − 432·47-s + 414·53-s + 648·55-s + 684·59-s + 422·61-s + 360·65-s − 332·67-s + 720·71-s + 26·73-s − 512·79-s − 2.37e3·83-s − 324·85-s + 630·89-s − 1.80e3·95-s + 2.10e3·97-s + ⋯ |
| L(s) = 1 | + 1.60·5-s + 0.986·11-s + 0.426·13-s − 0.256·17-s − 1.20·19-s + 0.652·23-s + 25-s + 2.99·29-s − 0.0926·31-s + 1.00·37-s + 0.685·41-s + 3.20·43-s − 1.34·47-s + 1.07·53-s + 1.58·55-s + 1.50·59-s + 0.885·61-s + 0.686·65-s − 0.605·67-s + 1.20·71-s + 0.0416·73-s − 0.729·79-s − 3.14·83-s − 0.413·85-s + 0.750·89-s − 1.94·95-s + 2.20·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.983345113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.983345113\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 18 T + 199 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 36 T - 35 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 10 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 18 T - 4589 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 100 T + 3141 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 72 T - 6983 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 234 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 16 T - 29535 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 226 T + 423 T^{2} - 226 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 90 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 452 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 432 T + 82801 T^{2} + 432 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 414 T + 22519 T^{2} - 414 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 684 T + 262477 T^{2} - 684 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 422 T - 48897 T^{2} - 422 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 332 T - 190539 T^{2} + 332 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 360 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 26 T - 388341 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 512 T - 230895 T^{2} + 512 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 1188 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 630 T - 308069 T^{2} - 630 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1054 T + p^{3} T^{2} )^{2} \) |
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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
−9.037960831042896969975298680303, −8.872689747802781363062763969872, −8.355825540664815877638017455014, −8.271076886378571913549930628921, −7.27405864455904072931175606736, −7.22266635615765697553398573165, −6.53918186103063353432003556766, −6.24187973631704266964349974091, −6.09921993341445797875673522651, −5.65188642908937928954359213343, −5.06782069272426223442837093718, −4.59248429739517556872411255422, −4.19164748514977124139850659722, −3.84795732219571888673878988638, −2.96128551253210483150595753623, −2.58795128738226286116707429988, −2.23071792738050317490231656841, −1.61247760154480089869499718296, −0.887450793374668581187595244892, −0.77059212253266433876616503756,
0.77059212253266433876616503756, 0.887450793374668581187595244892, 1.61247760154480089869499718296, 2.23071792738050317490231656841, 2.58795128738226286116707429988, 2.96128551253210483150595753623, 3.84795732219571888673878988638, 4.19164748514977124139850659722, 4.59248429739517556872411255422, 5.06782069272426223442837093718, 5.65188642908937928954359213343, 6.09921993341445797875673522651, 6.24187973631704266964349974091, 6.53918186103063353432003556766, 7.22266635615765697553398573165, 7.27405864455904072931175606736, 8.271076886378571913549930628921, 8.355825540664815877638017455014, 8.872689747802781363062763969872, 9.037960831042896969975298680303
