| 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
| \(L(1)\) |
\(\approx\) |
\(0.3346317037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3346317037\) |
| \(L(\frac{3}{2})\) |
|
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
−6.15428911739918075958513043457, −5.82578082042403876823314081078, −5.68455040462094995673312831711, −5.40988605961734331447787056191, −5.25667456811293638583315219380, −4.99795360547043055617557887558, −4.87099830372604334086163369089, −4.72517333060376803476830940823, −4.54389904671939813419331350747, −4.24727286286229418491598984029, −4.01684420068330271078517764083, −3.99937524080539103449705512100, −3.61171517275702277158679390960, −3.42658903803843866909974225979, −3.32400727565855169226455093682, −2.62436325602347059777791962325, −2.55617661194801175646635814555, −2.30990119237801905138515882506, −1.90976916441416359320012703533, −1.78531837338236419700659933121, −1.56599339567672057757819339034, −1.11819745092680352872338581209, −0.74819355755215240958031108992, −0.62115092589962802757037081904, −0.17966654048171228535908203735,
0.17966654048171228535908203735, 0.62115092589962802757037081904, 0.74819355755215240958031108992, 1.11819745092680352872338581209, 1.56599339567672057757819339034, 1.78531837338236419700659933121, 1.90976916441416359320012703533, 2.30990119237801905138515882506, 2.55617661194801175646635814555, 2.62436325602347059777791962325, 3.32400727565855169226455093682, 3.42658903803843866909974225979, 3.61171517275702277158679390960, 3.99937524080539103449705512100, 4.01684420068330271078517764083, 4.24727286286229418491598984029, 4.54389904671939813419331350747, 4.72517333060376803476830940823, 4.87099830372604334086163369089, 4.99795360547043055617557887558, 5.25667456811293638583315219380, 5.40988605961734331447787056191, 5.68455040462094995673312831711, 5.82578082042403876823314081078, 6.15428911739918075958513043457
