| L(s) = 1 | + 3-s + 5·5-s + 12·7-s − 2·9-s − 3·11-s − 13-s + 5·15-s + 3·17-s + 10·19-s + 12·21-s − 9·23-s + 10·25-s + 15·29-s − 3·31-s − 3·33-s + 60·35-s − 17·37-s − 39-s + 13·41-s + 16·43-s − 10·45-s − 23·47-s + 62·49-s + 3·51-s − 16·53-s − 15·55-s + 10·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 2.23·5-s + 4.53·7-s − 2/3·9-s − 0.904·11-s − 0.277·13-s + 1.29·15-s + 0.727·17-s + 2.29·19-s + 2.61·21-s − 1.87·23-s + 2·25-s + 2.78·29-s − 0.538·31-s − 0.522·33-s + 10.1·35-s − 2.79·37-s − 0.160·39-s + 2.03·41-s + 2.43·43-s − 1.49·45-s − 3.35·47-s + 62/7·49-s + 0.420·51-s − 2.19·53-s − 2.02·55-s + 1.32·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.205777517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.205777517\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_4$ | \( 1 - p T + 3 p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| good | 3 | $C_2^2:C_4$ | \( 1 - T + p T^{2} - 5 T^{3} + 16 T^{4} - 5 p T^{5} + p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 11 | $C_2^2:C_4$ | \( 1 + 3 T + 8 T^{2} + 51 T^{3} + 265 T^{4} + 51 p T^{5} + 8 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_4\times C_2$ | \( 1 + T - 12 T^{2} - 25 T^{3} + 131 T^{4} - 25 p T^{5} - 12 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 3 T + 37 T^{2} - 45 T^{3} + 676 T^{4} - 45 p T^{5} + 37 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 10 T + 21 T^{2} + 70 T^{3} - 469 T^{4} + 70 p T^{5} + 21 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 + 9 T + 8 T^{2} - 195 T^{3} - 1259 T^{4} - 195 p T^{5} + 8 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 - 15 T + 56 T^{2} + 315 T^{3} - 3629 T^{4} + 315 p T^{5} + 56 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 3 T - 12 T^{2} + 131 T^{3} + 1365 T^{4} + 131 p T^{5} - 12 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 17 T + 147 T^{2} + 1135 T^{3} + 7976 T^{4} + 1135 p T^{5} + 147 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 13 T + 28 T^{2} + 169 T^{3} - 945 T^{4} + 169 p T^{5} + 28 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 8 T + 97 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 23 T + 202 T^{2} + 925 T^{3} + 4101 T^{4} + 925 p T^{5} + 202 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 16 T + 43 T^{2} - 700 T^{3} - 7959 T^{4} - 700 p T^{5} + 43 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 - 10 T + T^{2} - 200 T^{3} + 5061 T^{4} - 200 p T^{5} + p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 2 T + 3 T^{2} + 424 T^{3} + 4265 T^{4} + 424 p T^{5} + 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 + 13 T + 12 T^{2} - 895 T^{3} - 9919 T^{4} - 895 p T^{5} + 12 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_4\times C_2$ | \( 1 + 3 T - 62 T^{2} - 399 T^{3} + 3205 T^{4} - 399 p T^{5} - 62 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 14 T + 63 T^{2} - 850 T^{3} + 12521 T^{4} - 850 p T^{5} + 63 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 10 T - 39 T^{2} - 10 p T^{3} - 2839 T^{4} - 10 p^{2} T^{5} - 39 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - T + 58 T^{2} - 335 T^{3} + 7041 T^{4} - 335 p T^{5} + 58 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_4\times C_2$ | \( 1 - 10 T - 29 T^{2} - 200 T^{3} + 10101 T^{4} - 200 p T^{5} - 29 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 22 T + 207 T^{2} + 2420 T^{3} + 31541 T^{4} + 2420 p T^{5} + 207 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.997254195665320768416219576909, −7.961595600537195664352958952845, −7.952299237160884627087736734550, −7.75269501983470168296675497588, −7.35850653024209585994170238437, −6.93521912190920406055813921907, −6.70161753087092963578358559930, −6.23180748413829489444945278014, −6.18207247947277812704345338023, −5.49686846298366543342347085464, −5.42161843309479855392362108319, −5.39413828176502001811686899506, −5.35278248576466047542935991219, −4.67342551298250568267422256156, −4.65785322453635944543271097618, −4.59539011608127370330086397284, −3.99851397107739463628333312448, −3.47526936478783636205290516813, −2.93036188926883865560803326429, −2.85297794061122481637129880291, −2.37283253929109365254223604174, −2.08217233944718027530155144274, −1.61160961171874541688535743557, −1.37281552333712775917710999015, −1.33740406821120752643164221842,
1.33740406821120752643164221842, 1.37281552333712775917710999015, 1.61160961171874541688535743557, 2.08217233944718027530155144274, 2.37283253929109365254223604174, 2.85297794061122481637129880291, 2.93036188926883865560803326429, 3.47526936478783636205290516813, 3.99851397107739463628333312448, 4.59539011608127370330086397284, 4.65785322453635944543271097618, 4.67342551298250568267422256156, 5.35278248576466047542935991219, 5.39413828176502001811686899506, 5.42161843309479855392362108319, 5.49686846298366543342347085464, 6.18207247947277812704345338023, 6.23180748413829489444945278014, 6.70161753087092963578358559930, 6.93521912190920406055813921907, 7.35850653024209585994170238437, 7.75269501983470168296675497588, 7.952299237160884627087736734550, 7.961595600537195664352958952845, 7.997254195665320768416219576909
