| L(s) = 1 | + 2·4-s − 32·13-s − 32·19-s − 32·25-s + 88·31-s + 68·37-s − 160·43-s − 64·52-s + 100·61-s − 8·64-s − 16·67-s − 32·73-s − 64·76-s + 152·79-s − 704·97-s − 64·100-s − 56·103-s − 112·109-s + 46·121-s + 176·124-s + 127-s + 131-s + 137-s + 139-s + 136·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 2.46·13-s − 1.68·19-s − 1.27·25-s + 2.83·31-s + 1.83·37-s − 3.72·43-s − 1.23·52-s + 1.63·61-s − 1/8·64-s − 0.238·67-s − 0.438·73-s − 0.842·76-s + 1.92·79-s − 7.25·97-s − 0.639·100-s − 0.543·103-s − 1.02·109-s + 0.380·121-s + 1.41·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.918·148-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.159294942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.159294942\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 + 32 T^{2} + 399 T^{4} + 32 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 75 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )( 1 + 14 T + 75 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 + 416 T^{2} + 89535 T^{4} + 416 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 16 T - 105 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 770 T^{2} + 313059 T^{4} + 770 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 1664 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 44 T + 975 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 34 T - 213 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 1184 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 2782 T^{2} + 2859843 T^{4} - 2782 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 4160 T^{2} + 9415119 T^{4} + 4160 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2^3$ | \( 1 + 5810 T^{2} + 21638739 T^{4} + 5810 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 50 T - 1221 T^{2} - 50 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 8 T - 4425 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 7490 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 16 T - 5073 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 76 T - 465 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 334 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 + 15680 T^{2} + 183120159 T^{4} + 15680 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2$ | \( ( 1 + 176 T + p^{2} T^{2} )^{4} \) |
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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
−6.88685179351601789470443518664, −6.83156574211931491978862277301, −6.71855948257917976708457849415, −6.36437199675942058731137304594, −6.30094403737312104757359129181, −5.99497647059754624686600982061, −5.75277862733876250615914647169, −5.21698595323224077019807140228, −5.11393037739498814138157043515, −5.05594913024561602012538896841, −4.88923233735178366573834220899, −4.23555280336039507462977398885, −4.22034388065030261400121565368, −3.99551639365702219500021050689, −3.96506829180250120192089407213, −3.18961281053659149247840428680, −2.85013092257451803809625240774, −2.76816409361014105210831278420, −2.70082817516649637689024406973, −2.07605373161724496090499518268, −2.04943702199489702666422573324, −1.59101870576369595937035838314, −1.23593945395795087787291779759, −0.39842264103706385416972307639, −0.39739311974999042794728534271,
0.39739311974999042794728534271, 0.39842264103706385416972307639, 1.23593945395795087787291779759, 1.59101870576369595937035838314, 2.04943702199489702666422573324, 2.07605373161724496090499518268, 2.70082817516649637689024406973, 2.76816409361014105210831278420, 2.85013092257451803809625240774, 3.18961281053659149247840428680, 3.96506829180250120192089407213, 3.99551639365702219500021050689, 4.22034388065030261400121565368, 4.23555280336039507462977398885, 4.88923233735178366573834220899, 5.05594913024561602012538896841, 5.11393037739498814138157043515, 5.21698595323224077019807140228, 5.75277862733876250615914647169, 5.99497647059754624686600982061, 6.30094403737312104757359129181, 6.36437199675942058731137304594, 6.71855948257917976708457849415, 6.83156574211931491978862277301, 6.88685179351601789470443518664
