| L(s) = 1 | + 6·5-s − 10·7-s − 33·13-s + 6·17-s − 26·19-s − 24·23-s + 25·25-s − 54·29-s − 60·35-s + 72·41-s + 25·43-s − 42·47-s − 23·49-s − 108·53-s − 126·59-s − 43·61-s − 198·65-s + 99·67-s − 108·71-s − 11·73-s + 3·79-s − 252·83-s + 36·85-s + 18·89-s + 330·91-s − 156·95-s − 228·97-s + ⋯ |
| L(s) = 1 | + 6/5·5-s − 1.42·7-s − 2.53·13-s + 6/17·17-s − 1.36·19-s − 1.04·23-s + 25-s − 1.86·29-s − 1.71·35-s + 1.75·41-s + 0.581·43-s − 0.893·47-s − 0.469·49-s − 2.03·53-s − 2.13·59-s − 0.704·61-s − 3.04·65-s + 1.47·67-s − 1.52·71-s − 0.150·73-s + 3/79·79-s − 3.03·83-s + 0.423·85-s + 0.202·89-s + 3.62·91-s − 1.64·95-s − 2.35·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.06464347841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06464347841\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 26 T + p^{2} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 6 T + 11 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 33 T + 532 T^{2} + 33 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6 T - 253 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 24 T + 47 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 54 T + 1813 T^{2} + 54 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 1055 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 937 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 72 T + 3409 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 25 T - 1224 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 42 T - 445 T^{2} + 42 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 108 T + 6697 T^{2} + 108 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 126 T + 8773 T^{2} + 126 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 43 T - 1872 T^{2} + 43 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 99 T + 7756 T^{2} - 99 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 108 T + 8929 T^{2} + 108 p^{2} T^{3} + p^{4} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 11 T - 5208 T^{2} + 11 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3 T + 6244 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 126 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 18 T + 8029 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 228 T + 26737 T^{2} + 228 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−10.56647384704009064982594692848, −9.881624878995777175377063863104, −9.579117491332307959902552706799, −9.367029836696207122995430480782, −9.197994054782095037874489469637, −8.233802395712900664747308924230, −7.88233313358470050031356328436, −7.28543250336424490121613075555, −7.04105015892287891270121439627, −6.28374200190273088852360084825, −6.15051146277458375813302244145, −5.67325995674424755030461103674, −5.09495717838486606591370473522, −4.48749437261910730339440740044, −4.11929292337683045969788036633, −3.00913667717193496210934945475, −2.94795065271375083421373374208, −2.08630420622431793551764499965, −1.71518724832366238203402832107, −0.082841613900554177401101565810,
0.082841613900554177401101565810, 1.71518724832366238203402832107, 2.08630420622431793551764499965, 2.94795065271375083421373374208, 3.00913667717193496210934945475, 4.11929292337683045969788036633, 4.48749437261910730339440740044, 5.09495717838486606591370473522, 5.67325995674424755030461103674, 6.15051146277458375813302244145, 6.28374200190273088852360084825, 7.04105015892287891270121439627, 7.28543250336424490121613075555, 7.88233313358470050031356328436, 8.233802395712900664747308924230, 9.197994054782095037874489469637, 9.367029836696207122995430480782, 9.579117491332307959902552706799, 9.881624878995777175377063863104, 10.56647384704009064982594692848
