| L(s) = 1 | + 2·2-s + 2·3-s + 4-s + 2·5-s + 4·6-s − 2·8-s + 9-s + 4·10-s − 4·11-s + 2·12-s + 4·15-s − 4·16-s + 6·17-s + 2·18-s − 12·19-s + 2·20-s − 8·22-s − 2·23-s − 4·24-s + 25-s − 2·27-s + 2·29-s + 8·30-s + 8·31-s − 2·32-s − 8·33-s + 12·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 1/2·4-s + 0.894·5-s + 1.63·6-s − 0.707·8-s + 1/3·9-s + 1.26·10-s − 1.20·11-s + 0.577·12-s + 1.03·15-s − 16-s + 1.45·17-s + 0.471·18-s − 2.75·19-s + 0.447·20-s − 1.70·22-s − 0.417·23-s − 0.816·24-s + 1/5·25-s − 0.384·27-s + 0.371·29-s + 1.46·30-s + 1.43·31-s − 0.353·32-s − 1.39·33-s + 2.05·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.422125789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.422125789\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 + 2 T - 5 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 6 T - 12 T^{3} + 395 T^{4} - 12 p T^{5} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2 T - 48 T^{2} + 12 T^{3} + 1747 T^{4} + 12 p T^{5} - 48 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 6 T + 55 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 33 T^{2} - 348 T^{3} + 4736 T^{4} - 348 p T^{5} + 33 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 6 T - 7 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 4 T + 30 T^{2} - 432 T^{3} - 2765 T^{4} - 432 p T^{5} + 30 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 4 T - 87 T^{2} + 12 T^{3} + 6952 T^{4} + 12 p T^{5} - 87 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 90 T^{2} + 4619 T^{4} - 90 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 2 T - 56 T^{2} + 124 T^{3} - 365 T^{4} + 124 p T^{5} - 56 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 14 T + 20 T^{2} - 588 T^{3} + 13355 T^{4} - 588 p T^{5} + 20 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 22 T + 228 T^{2} - 2508 T^{3} + 26131 T^{4} - 2508 p T^{5} + 228 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 22 T + 224 T^{2} + 2508 T^{3} + 26939 T^{4} + 2508 p T^{5} + 224 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^3$ | \( 1 - 130 T^{2} + 10659 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 3 T^{2} - 7912 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 2 T - 184 T^{2} - 12 T^{3} + 25547 T^{4} - 12 p T^{5} - 184 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.88515414424776816259659230251, −7.66543134650304003649107453855, −7.33300887260345834928555608050, −6.79235315383597079819646683844, −6.62687841902815410737963705239, −6.45611960966774423908519883295, −6.38030463779941462824574333830, −5.83051844556322620255475760146, −5.82907693754770665699820378681, −5.54035984744261608199304680853, −5.40508462689383278049719813251, −4.68316212317298898606494462527, −4.67111069455917670056302021452, −4.66894962608126442547766659051, −4.39963802203713470232055659529, −3.78739608329031319221942992866, −3.54789851137010996776476931703, −3.47258175047305593146733615536, −3.07140812508194113775194924889, −2.60320832183567795665031102675, −2.48870576477034165880532921844, −2.22423637587724817772229916410, −1.94926380394777601894782614590, −1.26267219733485229774099777472, −0.54406922090901035924224570178,
0.54406922090901035924224570178, 1.26267219733485229774099777472, 1.94926380394777601894782614590, 2.22423637587724817772229916410, 2.48870576477034165880532921844, 2.60320832183567795665031102675, 3.07140812508194113775194924889, 3.47258175047305593146733615536, 3.54789851137010996776476931703, 3.78739608329031319221942992866, 4.39963802203713470232055659529, 4.66894962608126442547766659051, 4.67111069455917670056302021452, 4.68316212317298898606494462527, 5.40508462689383278049719813251, 5.54035984744261608199304680853, 5.82907693754770665699820378681, 5.83051844556322620255475760146, 6.38030463779941462824574333830, 6.45611960966774423908519883295, 6.62687841902815410737963705239, 6.79235315383597079819646683844, 7.33300887260345834928555608050, 7.66543134650304003649107453855, 7.88515414424776816259659230251
