| L(s) = 1 | − 4·2-s + 4·3-s + 10·4-s + 4·5-s − 16·6-s − 7-s − 20·8-s + 10·9-s − 16·10-s + 40·12-s − 4·13-s + 4·14-s + 16·15-s + 35·16-s + 5·17-s − 40·18-s − 7·19-s + 40·20-s − 4·21-s + 8·23-s − 80·24-s + 10·25-s + 16·26-s + 20·27-s − 10·28-s − 29-s − 64·30-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 2.30·3-s + 5·4-s + 1.78·5-s − 6.53·6-s − 0.377·7-s − 7.07·8-s + 10/3·9-s − 5.05·10-s + 11.5·12-s − 1.10·13-s + 1.06·14-s + 4.13·15-s + 35/4·16-s + 1.21·17-s − 9.42·18-s − 1.60·19-s + 8.94·20-s − 0.872·21-s + 1.66·23-s − 16.3·24-s + 2·25-s + 3.13·26-s + 3.84·27-s − 1.88·28-s − 0.185·29-s − 11.6·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.35844158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.35844158\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 3 | $C_1$ | \( ( 1 - T )^{4} \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 11 | | \( 1 \) |
| good | 7 | $C_2 \wr C_2\wr C_2$ | \( 1 + T + p T^{2} + 11 T^{3} + 100 T^{4} + 11 p T^{5} + p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 25 T^{2} + 44 T^{3} + 301 T^{4} + 44 p T^{5} + 25 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 - 5 T + 45 T^{2} - 155 T^{3} + 968 T^{4} - 155 p T^{5} + 45 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 + 7 T + 3 p T^{2} + 11 p T^{3} + 1224 T^{4} + 11 p^{2} T^{5} + 3 p^{3} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 63 T^{2} - 222 T^{3} + 1235 T^{4} - 222 p T^{5} + 63 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 + T + 95 T^{2} + 77 T^{3} + 3928 T^{4} + 77 p T^{5} + 95 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 - 9 T + 103 T^{2} - 647 T^{3} + 4244 T^{4} - 647 p T^{5} + 103 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 11 T + 93 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 3 T + 63 T^{2} + 211 T^{3} + 4400 T^{4} + 211 p T^{5} + 63 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 - T + 65 T^{2} - 371 T^{3} + 2296 T^{4} - 371 p T^{5} + 65 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 - 2 T + 37 T^{2} - 130 T^{3} + 1711 T^{4} - 130 p T^{5} + 37 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 - 5 T + 125 T^{2} - 555 T^{3} + 8228 T^{4} - 555 p T^{5} + 125 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 205 T^{2} - 2072 T^{3} + 17173 T^{4} - 2072 p T^{5} + 205 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 - 14 T + 232 T^{2} - 2130 T^{3} + 21566 T^{4} - 2130 p T^{5} + 232 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 - 29 T + 351 T^{2} - 2167 T^{3} + 11580 T^{4} - 2167 p T^{5} + 351 p^{2} T^{6} - 29 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 168 T^{2} + 1118 T^{3} + 15374 T^{4} + 1118 p T^{5} + 168 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 + 3 T + 107 T^{2} + 851 T^{3} + 10044 T^{4} + 851 p T^{5} + 107 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 - 5 T + 211 T^{2} - 985 T^{3} + 22216 T^{4} - 985 p T^{5} + 211 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 - 30 T + 580 T^{2} - 8210 T^{3} + 88598 T^{4} - 8210 p T^{5} + 580 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.37856459335755629157000476959, −5.86240335268329111813354578157, −5.85637220447766366951203618735, −5.57780076198255618771927649782, −5.43966913883498930110174411269, −4.88152951962199044387497456399, −4.71916645637676499641067974246, −4.71188221064685597850979841091, −4.68978772083149847956054285484, −3.93351556969401226757587831173, −3.74823117076989445935330995132, −3.56139946786399743164250102325, −3.52160548744005596348362890616, −3.01559758226567711089938181440, −2.86157182761029892584425456922, −2.73066706021954667986697265055, −2.42563515821397385390859168681, −2.19404113899020358772371896376, −2.11934066981889291897934828967, −1.93705153450468391415795383331, −1.93274471037415795064004195451, −0.990220550747843734830902737325, −0.934104173103604681196392331456, −0.849797425940650697161836309328, −0.69806775574472982221494687111,
0.69806775574472982221494687111, 0.849797425940650697161836309328, 0.934104173103604681196392331456, 0.990220550747843734830902737325, 1.93274471037415795064004195451, 1.93705153450468391415795383331, 2.11934066981889291897934828967, 2.19404113899020358772371896376, 2.42563515821397385390859168681, 2.73066706021954667986697265055, 2.86157182761029892584425456922, 3.01559758226567711089938181440, 3.52160548744005596348362890616, 3.56139946786399743164250102325, 3.74823117076989445935330995132, 3.93351556969401226757587831173, 4.68978772083149847956054285484, 4.71188221064685597850979841091, 4.71916645637676499641067974246, 4.88152951962199044387497456399, 5.43966913883498930110174411269, 5.57780076198255618771927649782, 5.85637220447766366951203618735, 5.86240335268329111813354578157, 6.37856459335755629157000476959
