| L(s) = 1 | + 4·2-s + 4·3-s + 10·4-s + 3·5-s + 16·6-s + 20·8-s + 10·9-s + 12·10-s + 2·11-s + 40·12-s + 5·13-s + 12·15-s + 35·16-s + 10·17-s + 40·18-s + 8·19-s + 30·20-s + 8·22-s − 4·23-s + 80·24-s + 25-s + 20·26-s + 20·27-s + 3·29-s + 48·30-s − 6·31-s + 56·32-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 2.30·3-s + 5·4-s + 1.34·5-s + 6.53·6-s + 7.07·8-s + 10/3·9-s + 3.79·10-s + 0.603·11-s + 11.5·12-s + 1.38·13-s + 3.09·15-s + 35/4·16-s + 2.42·17-s + 9.42·18-s + 1.83·19-s + 6.70·20-s + 1.70·22-s − 0.834·23-s + 16.3·24-s + 1/5·25-s + 3.92·26-s + 3.84·27-s + 0.557·29-s + 8.76·30-s − 1.07·31-s + 9.89·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(509.3357019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(509.3357019\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 - 3 T + 8 T^{2} - 13 T^{3} + 14 T^{4} - 13 p T^{5} + 8 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 2 T + 20 T^{2} - 6 p T^{3} + 230 T^{4} - 6 p^{2} T^{5} + 20 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 5 T + 46 T^{2} - 175 T^{3} + 850 T^{4} - 175 p T^{5} + 46 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 10 T + 64 T^{2} - 326 T^{3} + 1534 T^{4} - 326 p T^{5} + 64 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 3 T + 10 T^{2} - 105 T^{3} + 1642 T^{4} - 105 p T^{5} + 10 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 6 T + 32 T^{2} + 262 T^{3} + 2526 T^{4} + 262 p T^{5} + 32 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - T + 2 p T^{2} + 205 T^{3} + 2410 T^{4} + 205 p T^{5} + 2 p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - T + 84 T^{2} + 5 T^{3} + 4550 T^{4} + 5 p T^{5} + 84 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + T + 44 T^{2} + 9 T^{3} + 3558 T^{4} + 9 p T^{5} + 44 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 17 T + 186 T^{2} - 1449 T^{3} + 10610 T^{4} - 1449 p T^{5} + 186 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 4 T + 124 T^{2} - 92 T^{3} + 6870 T^{4} - 92 p T^{5} + 124 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 12 T^{2} - 128 T^{3} + 5462 T^{4} - 128 p T^{5} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 24 T + 404 T^{2} - 4600 T^{3} + 41910 T^{4} - 4600 p T^{5} + 404 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 8 T + 164 T^{2} + 440 T^{3} + 10182 T^{4} + 440 p T^{5} + 164 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 4 T + 124 T^{2} + 212 T^{3} + 11110 T^{4} + 212 p T^{5} + 124 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 6 T + 156 T^{2} - 834 T^{3} + 11990 T^{4} - 834 p T^{5} + 156 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 6 T + 180 T^{2} - 942 T^{3} + 15830 T^{4} - 942 p T^{5} + 180 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 18 T + 320 T^{2} - 3226 T^{3} + 34958 T^{4} - 3226 p T^{5} + 320 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 26 T + 568 T^{2} - 7462 T^{3} + 85294 T^{4} - 7462 p T^{5} + 568 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 13 T + 42 T^{2} + 1165 T^{3} - 14182 T^{4} + 1165 p T^{5} + 42 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.63144640075807542201263483903, −5.18615653881699095855959512480, −5.15567566533791267208171543070, −5.09069544221993695910682606452, −5.05620039834147318610565943564, −4.67040245985694514077494029664, −4.32243764665577772474847159405, −4.03834245033923570215542615508, −3.93790669836471316118870538039, −3.90191286145674244954038286533, −3.67673975106931476299694696671, −3.45142232743676817976483689097, −3.38118957467883120114702822282, −3.22758125657831922146054467816, −3.04541142594551000008738126202, −2.63585755395282678018770221494, −2.50622236186389871823298359303, −2.29358038621653405491030038926, −2.16493599201513966600051225527, −1.94406574855509788649892484348, −1.74761289676028603141045239165, −1.27196765750633436417331769890, −1.14518776653151587738100845789, −0.997677632990168216434250776389, −0.822450730284461916401498395376,
0.822450730284461916401498395376, 0.997677632990168216434250776389, 1.14518776653151587738100845789, 1.27196765750633436417331769890, 1.74761289676028603141045239165, 1.94406574855509788649892484348, 2.16493599201513966600051225527, 2.29358038621653405491030038926, 2.50622236186389871823298359303, 2.63585755395282678018770221494, 3.04541142594551000008738126202, 3.22758125657831922146054467816, 3.38118957467883120114702822282, 3.45142232743676817976483689097, 3.67673975106931476299694696671, 3.90191286145674244954038286533, 3.93790669836471316118870538039, 4.03834245033923570215542615508, 4.32243764665577772474847159405, 4.67040245985694514077494029664, 5.05620039834147318610565943564, 5.09069544221993695910682606452, 5.15567566533791267208171543070, 5.18615653881699095855959512480, 5.63144640075807542201263483903
