| L(s) = 1 | + 2-s + 4·3-s + 4·6-s + 10·9-s + 8·11-s + 7·13-s + 16-s + 6·17-s + 10·18-s − 3·19-s + 8·22-s + 2·23-s + 7·26-s + 20·27-s + 16·29-s − 9·31-s + 3·32-s + 32·33-s + 6·34-s + 8·37-s − 3·38-s + 28·39-s − 4·41-s + 5·43-s + 2·46-s − 6·47-s + 4·48-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 2.30·3-s + 1.63·6-s + 10/3·9-s + 2.41·11-s + 1.94·13-s + 1/4·16-s + 1.45·17-s + 2.35·18-s − 0.688·19-s + 1.70·22-s + 0.417·23-s + 1.37·26-s + 3.84·27-s + 2.97·29-s − 1.61·31-s + 0.530·32-s + 5.57·33-s + 1.02·34-s + 1.31·37-s − 0.486·38-s + 4.48·39-s − 0.624·41-s + 0.762·43-s + 0.294·46-s − 0.875·47-s + 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(58.42669546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(58.42669546\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 - T + T^{2} - T^{3} - p T^{5} + p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 8 T + 36 T^{2} - 84 T^{3} + 222 T^{4} - 84 p T^{5} + 36 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 7 T + 33 T^{2} - 94 T^{3} + 326 T^{4} - 94 p T^{5} + 33 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 6 T + 48 T^{2} - 198 T^{3} + 1094 T^{4} - 198 p T^{5} + 48 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 3 T + 39 T^{2} - 4 T^{3} + 564 T^{4} - 4 p T^{5} + 39 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 2 T + 24 T^{2} - 66 T^{3} + 1006 T^{4} - 66 p T^{5} + 24 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 9 T + 68 T^{2} + 261 T^{3} + 1526 T^{4} + 261 p T^{5} + 68 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 8 T + 86 T^{2} - 744 T^{3} + 4399 T^{4} - 744 p T^{5} + 86 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 4 T + 104 T^{2} + 256 T^{3} + 4966 T^{4} + 256 p T^{5} + 104 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 5 T + 120 T^{2} - 677 T^{3} + 6782 T^{4} - 677 p T^{5} + 120 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 6 T + 124 T^{2} + 346 T^{3} + 6382 T^{4} + 346 p T^{5} + 124 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 6 T + 136 T^{2} + 662 T^{3} + 9910 T^{4} + 662 p T^{5} + 136 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 10 T + 196 T^{2} + 1726 T^{3} + 16158 T^{4} + 1726 p T^{5} + 196 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - T + 251 T^{2} - 192 T^{3} + 24716 T^{4} - 192 p T^{5} + 251 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 22 T + 252 T^{2} - 1806 T^{3} + 12966 T^{4} - 1806 p T^{5} + 252 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 4 T + 170 T^{2} - 48 T^{3} + 12595 T^{4} - 48 p T^{5} + 170 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 8 T + 246 T^{2} - 1764 T^{3} + 26539 T^{4} - 1764 p T^{5} + 246 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 2 T + 220 T^{2} + 894 T^{3} + 22430 T^{4} + 894 p T^{5} + 220 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 10 T + 360 T^{2} - 2586 T^{3} + 48262 T^{4} - 2586 p T^{5} + 360 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 12 T + 330 T^{2} - 3296 T^{3} + 45123 T^{4} - 3296 p T^{5} + 330 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.05300246637730156897465135846, −5.97076253723510005420973136853, −5.45481547952287991989216434521, −5.45348843056417261321843581638, −5.35969805260832104299437542366, −4.72049261048357737871157543961, −4.64360357990520851089527022522, −4.57458359610688997249813230724, −4.46620191903369851979615133478, −3.98970806971868535235730001616, −3.92006814672219499741455425191, −3.75291665458010749580400992441, −3.53933961151873170294097159457, −3.36977852526509252181356361958, −3.26084662193463951224756427769, −3.03754690436243456373126155650, −2.76979021000270659381272254049, −2.35715009430686928701150760371, −2.27016349358226588116345036128, −1.88962634918161138770559425348, −1.57148602563313962427935096686, −1.52416719668915313522010323302, −0.969529348848911043312022142393, −0.895063884051016429995138867362, −0.78593594889558566311843536902,
0.78593594889558566311843536902, 0.895063884051016429995138867362, 0.969529348848911043312022142393, 1.52416719668915313522010323302, 1.57148602563313962427935096686, 1.88962634918161138770559425348, 2.27016349358226588116345036128, 2.35715009430686928701150760371, 2.76979021000270659381272254049, 3.03754690436243456373126155650, 3.26084662193463951224756427769, 3.36977852526509252181356361958, 3.53933961151873170294097159457, 3.75291665458010749580400992441, 3.92006814672219499741455425191, 3.98970806971868535235730001616, 4.46620191903369851979615133478, 4.57458359610688997249813230724, 4.64360357990520851089527022522, 4.72049261048357737871157543961, 5.35969805260832104299437542366, 5.45348843056417261321843581638, 5.45481547952287991989216434521, 5.97076253723510005420973136853, 6.05300246637730156897465135846
