| L(s) = 1 | + 2-s − 4·3-s + 3·5-s − 4·6-s + 10·9-s + 3·10-s − 2·11-s − 4·13-s − 12·15-s − 16-s + 2·17-s + 10·18-s − 7·19-s − 2·22-s + 3·23-s − 25-s − 4·26-s − 20·27-s + 29-s − 12·30-s − 3·31-s − 32-s + 8·33-s + 2·34-s + 10·37-s − 7·38-s + 16·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 2.30·3-s + 1.34·5-s − 1.63·6-s + 10/3·9-s + 0.948·10-s − 0.603·11-s − 1.10·13-s − 3.09·15-s − 1/4·16-s + 0.485·17-s + 2.35·18-s − 1.60·19-s − 0.426·22-s + 0.625·23-s − 1/5·25-s − 0.784·26-s − 3.84·27-s + 0.185·29-s − 2.19·30-s − 0.538·31-s − 0.176·32-s + 1.39·33-s + 0.342·34-s + 1.64·37-s − 1.13·38-s + 2.56·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.813898125\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.813898125\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 - T + T^{2} - T^{3} + p T^{4} - p T^{5} + p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 3 T + 2 p T^{2} - p^{2} T^{3} + 74 T^{4} - p^{3} T^{5} + 2 p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 2 T + 20 T^{2} + 34 T^{3} + 294 T^{4} + 34 p T^{5} + 20 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 2 T + 40 T^{2} - 62 T^{3} + 878 T^{4} - 62 p T^{5} + 40 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 7 T + 64 T^{2} + 351 T^{3} + 1774 T^{4} + 351 p T^{5} + 64 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 3 T + 40 T^{2} + 49 T^{3} + 494 T^{4} + 49 p T^{5} + 40 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - T + 86 T^{2} - 35 T^{3} + 3378 T^{4} - 35 p T^{5} + 86 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 3 T - 4 T^{2} + 119 T^{3} + 58 p T^{4} + 119 p T^{5} - 4 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 10 T + 64 T^{2} - 270 T^{3} + 1870 T^{4} - 270 p T^{5} + 64 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 16 T + 4 p T^{2} - 1280 T^{3} + 8694 T^{4} - 1280 p T^{5} + 4 p^{3} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 3 T + 128 T^{2} - 275 T^{3} + 7246 T^{4} - 275 p T^{5} + 128 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 5 T + 148 T^{2} + 689 T^{3} + 9638 T^{4} + 689 p T^{5} + 148 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 5 T + 174 T^{2} - 727 T^{3} + 12802 T^{4} - 727 p T^{5} + 174 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 20 T + 316 T^{2} - 3236 T^{3} + 28790 T^{4} - 3236 p T^{5} + 316 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 12 T + 180 T^{2} + 1508 T^{3} + 15014 T^{4} + 1508 p T^{5} + 180 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 22 T + 228 T^{2} + 1254 T^{3} + 6086 T^{4} + 1254 p T^{5} + 228 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 52 T^{2} + 304 T^{3} + 7478 T^{4} + 304 p T^{5} + 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 13 T + 126 T^{2} + 261 T^{3} - 3934 T^{4} + 261 p T^{5} + 126 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 11 T + 196 T^{2} - 1167 T^{3} + 15030 T^{4} - 1167 p T^{5} + 196 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + T + 296 T^{2} + 329 T^{3} + 35310 T^{4} + 329 p T^{5} + 296 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 5 T + 194 T^{2} - 139 T^{3} + 16986 T^{4} - 139 p T^{5} + 194 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 17 T + 374 T^{2} - 4127 T^{3} + 52210 T^{4} - 4127 p T^{5} + 374 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.32560845913887841297471198901, −6.30901291294132636685115970855, −6.10499566907420669336538543496, −5.91746598278447593763807550948, −5.83559009690674376137725567846, −5.50077887191834548338236696429, −5.08642110563666370220060422999, −5.06719818170744336786515526787, −5.06518049332531632072665421331, −4.79564159119713193983318154562, −4.41033295839229128740034148081, −4.29445653647676141247455766471, −4.16439834897868146107249918990, −3.80335383836124672653325839043, −3.61099934526011828558810157583, −3.25830501722698521304745116018, −2.71992528217226897427403486343, −2.53705785222885507365364174128, −2.49963374130400013679772168831, −2.02028293195948253397640087987, −1.82594833322199503062431996154, −1.57733661967134253562589763215, −0.845168618162541683609612433952, −0.816516820932596110321388874134, −0.31943452081985678328832852986,
0.31943452081985678328832852986, 0.816516820932596110321388874134, 0.845168618162541683609612433952, 1.57733661967134253562589763215, 1.82594833322199503062431996154, 2.02028293195948253397640087987, 2.49963374130400013679772168831, 2.53705785222885507365364174128, 2.71992528217226897427403486343, 3.25830501722698521304745116018, 3.61099934526011828558810157583, 3.80335383836124672653325839043, 4.16439834897868146107249918990, 4.29445653647676141247455766471, 4.41033295839229128740034148081, 4.79564159119713193983318154562, 5.06518049332531632072665421331, 5.06719818170744336786515526787, 5.08642110563666370220060422999, 5.50077887191834548338236696429, 5.83559009690674376137725567846, 5.91746598278447593763807550948, 6.10499566907420669336538543496, 6.30901291294132636685115970855, 6.32560845913887841297471198901
