| L(s) = 1 | − 8·3-s + 30·9-s + 80·19-s + 68·25-s − 40·27-s + 144·43-s − 36·49-s − 640·57-s + 176·67-s + 200·73-s − 544·75-s − 205·81-s + 200·97-s + 444·121-s + 127-s − 1.15e3·129-s + 131-s + 137-s + 139-s + 288·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 36·169-s + 2.40e3·171-s + ⋯ |
| L(s) = 1 | − 8/3·3-s + 10/3·9-s + 4.21·19-s + 2.71·25-s − 1.48·27-s + 3.34·43-s − 0.734·49-s − 11.2·57-s + 2.62·67-s + 2.73·73-s − 7.25·75-s − 2.53·81-s + 2.06·97-s + 3.66·121-s + 0.00787·127-s − 8.93·129-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 1.95·147-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.213·169-s + 14.0·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.721789652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.721789652\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 222 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 258 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 20 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 802 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 1022 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 1202 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2082 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 36 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 322 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 5218 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 3618 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 7122 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 44 T + p^{2} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 3682 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 6002 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 3198 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 10078 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.34012045453694578741428696917, −6.80288214578388491746736169686, −6.65364591297747870710388689786, −6.42820932354745428256118555880, −6.33516711498396346401480547295, −6.01221191502625621435231793013, −5.71677700872871977321131061401, −5.42500492082428128920810935536, −5.41421042221185221915126677044, −5.08600164878007772042224838467, −4.92444239602724890589805999927, −4.81427986004129152535170575064, −4.65275896454089399583749313423, −3.95131935146338699545799978255, −3.87128202907292300840031248089, −3.54527821938954884458569596526, −3.21073990467791063163540115425, −2.80788580915471038117900769069, −2.70522866401765373102466150185, −2.32752055279160290121155034862, −1.64308797259786427134807508849, −1.07902397741686234209396157843, −1.01589874925361876581605265108, −0.75717236531517540351762108381, −0.49638968104150882423157089140,
0.49638968104150882423157089140, 0.75717236531517540351762108381, 1.01589874925361876581605265108, 1.07902397741686234209396157843, 1.64308797259786427134807508849, 2.32752055279160290121155034862, 2.70522866401765373102466150185, 2.80788580915471038117900769069, 3.21073990467791063163540115425, 3.54527821938954884458569596526, 3.87128202907292300840031248089, 3.95131935146338699545799978255, 4.65275896454089399583749313423, 4.81427986004129152535170575064, 4.92444239602724890589805999927, 5.08600164878007772042224838467, 5.41421042221185221915126677044, 5.42500492082428128920810935536, 5.71677700872871977321131061401, 6.01221191502625621435231793013, 6.33516711498396346401480547295, 6.42820932354745428256118555880, 6.65364591297747870710388689786, 6.80288214578388491746736169686, 7.34012045453694578741428696917
