| L(s) = 1 | + 4·2-s + 4·3-s + 10·4-s − 5-s + 16·6-s + 2·7-s + 20·8-s + 10·9-s − 4·10-s + 15·11-s + 40·12-s + 11·13-s + 8·14-s − 4·15-s + 35·16-s + 2·17-s + 40·18-s − 2·19-s − 10·20-s + 8·21-s + 60·22-s − 4·23-s + 80·24-s − 10·25-s + 44·26-s + 20·27-s + 20·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 2.30·3-s + 5·4-s − 0.447·5-s + 6.53·6-s + 0.755·7-s + 7.07·8-s + 10/3·9-s − 1.26·10-s + 4.52·11-s + 11.5·12-s + 3.05·13-s + 2.13·14-s − 1.03·15-s + 35/4·16-s + 0.485·17-s + 9.42·18-s − 0.458·19-s − 2.23·20-s + 1.74·21-s + 12.7·22-s − 0.834·23-s + 16.3·24-s − 2·25-s + 8.62·26-s + 3.84·27-s + 3.77·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 23^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 23^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(354.5865738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(354.5865738\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 23 | $C_1$ | \( ( 1 + T )^{4} \) |
| 29 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 + T + 11 T^{2} + 4 p T^{3} + 64 T^{4} + 4 p^{2} T^{5} + 11 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 2 T + 16 T^{2} - 22 T^{3} + 142 T^{4} - 22 p T^{5} + 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 15 T + p^{2} T^{2} - 648 T^{3} + 2512 T^{4} - 648 p T^{5} + p^{4} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 11 T + 81 T^{2} - 430 T^{3} + 1730 T^{4} - 430 p T^{5} + 81 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 2 T + 56 T^{2} - 82 T^{3} + 1342 T^{4} - 82 p T^{5} + 56 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 2 T + 64 T^{2} + 94 T^{3} + 1726 T^{4} + 94 p T^{5} + 64 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 5 T + 91 T^{2} + 324 T^{3} + 116 p T^{4} + 324 p T^{5} + 91 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 3 T + 71 T^{2} - 54 T^{3} + 2842 T^{4} - 54 p T^{5} + 71 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 11 T + 153 T^{2} - 1330 T^{3} + 9186 T^{4} - 1330 p T^{5} + 153 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 10 T + 144 T^{2} - 962 T^{3} + 8270 T^{4} - 962 p T^{5} + 144 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 10 T + 100 T^{2} - 278 T^{3} + 2646 T^{4} - 278 p T^{5} + 100 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 24 T + 248 T^{2} - 1316 T^{3} + 118 p T^{4} - 1316 p T^{5} + 248 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - T + 107 T^{2} - 360 T^{3} + 6260 T^{4} - 360 p T^{5} + 107 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 3 T + 79 T^{2} - 166 T^{3} + 7406 T^{4} - 166 p T^{5} + 79 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 7 T + 125 T^{2} + 210 T^{3} + 3450 T^{4} + 210 p T^{5} + 125 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 13 T + 247 T^{2} + 2036 T^{3} + 24016 T^{4} + 2036 p T^{5} + 247 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 2 T + 140 T^{2} + 474 T^{3} + 12990 T^{4} + 474 p T^{5} + 140 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 22 T + 396 T^{2} + 4322 T^{3} + 45006 T^{4} + 4322 p T^{5} + 396 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 16 T + 276 T^{2} + 2576 T^{3} + 29814 T^{4} + 2576 p T^{5} + 276 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 8 T + 200 T^{2} + 244 T^{3} + 14638 T^{4} + 244 p T^{5} + 200 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 4 T + 184 T^{2} + 944 T^{3} + 25342 T^{4} + 944 p T^{5} + 184 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.03551395824802504828667213189, −5.68078310836492675432732822605, −5.61337307976792367476115288820, −5.51022996749500273383288064181, −5.25748536513184171561118047583, −4.54124433259298774656434471527, −4.35569426831636427665935057582, −4.35280701315043342251846956946, −4.34233859308691719431140050594, −4.09580146021219417373190717410, −3.93599366292903357364614241732, −3.84923644699699218054776427774, −3.72083046823422741114440948161, −3.41425895700352252283425616021, −3.18877525123243432882227308619, −3.05465417335426516201739838248, −2.97527158551406448730150668195, −2.26911511414332033860136081135, −2.05540283793924472189019933892, −2.04020497489141797281950654138, −1.99368217528527194687172662846, −1.29361778423481588234395321698, −1.22594867015355268222048530624, −1.08785004485030173178600087229, −1.01791186376631129631626723294,
1.01791186376631129631626723294, 1.08785004485030173178600087229, 1.22594867015355268222048530624, 1.29361778423481588234395321698, 1.99368217528527194687172662846, 2.04020497489141797281950654138, 2.05540283793924472189019933892, 2.26911511414332033860136081135, 2.97527158551406448730150668195, 3.05465417335426516201739838248, 3.18877525123243432882227308619, 3.41425895700352252283425616021, 3.72083046823422741114440948161, 3.84923644699699218054776427774, 3.93599366292903357364614241732, 4.09580146021219417373190717410, 4.34233859308691719431140050594, 4.35280701315043342251846956946, 4.35569426831636427665935057582, 4.54124433259298774656434471527, 5.25748536513184171561118047583, 5.51022996749500273383288064181, 5.61337307976792367476115288820, 5.68078310836492675432732822605, 6.03551395824802504828667213189
