| L(s) = 1 | − 4·3-s + 2·5-s − 4·7-s + 10·9-s − 3·11-s + 4·13-s − 8·15-s + 3·17-s − 4·19-s + 16·21-s − 8·23-s − 25-s − 20·27-s + 4·29-s + 9·31-s + 12·33-s − 8·35-s + 3·37-s − 16·39-s + 6·41-s − 8·43-s + 20·45-s + 15·47-s + 10·49-s − 12·51-s + 13·53-s − 6·55-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 0.894·5-s − 1.51·7-s + 10/3·9-s − 0.904·11-s + 1.10·13-s − 2.06·15-s + 0.727·17-s − 0.917·19-s + 3.49·21-s − 1.66·23-s − 1/5·25-s − 3.84·27-s + 0.742·29-s + 1.61·31-s + 2.08·33-s − 1.35·35-s + 0.493·37-s − 2.56·39-s + 0.937·41-s − 1.21·43-s + 2.98·45-s + 2.18·47-s + 10/7·49-s − 1.68·51-s + 1.78·53-s − 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.535329846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.535329846\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 - 2 T + p T^{2} - 6 T^{3} + 4 T^{4} - 6 p T^{5} + p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 3 T + 10 T^{2} - 13 T^{3} + 26 T^{4} - 13 p T^{5} + 10 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 3 T + 32 T^{2} - 133 T^{3} + 526 T^{4} - 133 p T^{5} + 32 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 4 T + 31 T^{2} + 68 T^{3} + 440 T^{4} + 68 p T^{5} + 31 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 8 T + 51 T^{2} + 320 T^{3} + 2040 T^{4} + 320 p T^{5} + 51 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 4 T + 71 T^{2} - 312 T^{3} + 2544 T^{4} - 312 p T^{5} + 71 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 9 T + 92 T^{2} - 613 T^{3} + 3526 T^{4} - 613 p T^{5} + 92 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 3 T + 112 T^{2} - 313 T^{3} + 5566 T^{4} - 313 p T^{5} + 112 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 6 T + 136 T^{2} - 602 T^{3} + 7854 T^{4} - 602 p T^{5} + 136 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 8 T + 131 T^{2} + 800 T^{3} + 8320 T^{4} + 800 p T^{5} + 131 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 15 T + 232 T^{2} - 1931 T^{3} + 16622 T^{4} - 1931 p T^{5} + 232 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 13 T + 70 T^{2} - 463 T^{3} + 4946 T^{4} - 463 p T^{5} + 70 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 8 T + 148 T^{2} - 1064 T^{3} + 12806 T^{4} - 1064 p T^{5} + 148 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 9 T + 14 T^{2} + 197 T^{3} + 322 T^{4} + 197 p T^{5} + 14 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 8 T^{2} - 32 T^{3} + 8158 T^{4} - 32 p T^{5} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 8 T + 196 T^{2} + 1352 T^{3} + 20054 T^{4} + 1352 p T^{5} + 196 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 32 T + 611 T^{2} - 8012 T^{3} + 79456 T^{4} - 8012 p T^{5} + 611 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + T + 198 T^{2} - 387 T^{3} + 17970 T^{4} - 387 p T^{5} + 198 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 13 T + 254 T^{2} - 2165 T^{3} + 28530 T^{4} - 2165 p T^{5} + 254 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 7 T + 334 T^{2} - 1657 T^{3} + 43282 T^{4} - 1657 p T^{5} + 334 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 19 T + 318 T^{2} - 4413 T^{3} + 47202 T^{4} - 4413 p T^{5} + 318 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.50787008838141002125419353245, −6.02016552987791367905355228044, −6.01702274126565034885151224811, −5.97711329369706833411246555965, −5.69304113418518353496041083509, −5.45195373506211578202097913681, −5.28583680920974565499312676872, −5.17687241990234323073743359119, −4.85919671152643767163723918939, −4.45331343981079139193833447446, −4.27940793419390527658207975584, −4.15731785954268784274056877998, −4.12919432357728490153279108294, −3.41735063238555423679647488311, −3.40135721863220317049830975014, −3.39128726755421323402875748274, −2.91223682695815429669506877567, −2.26242807367851098693190799534, −2.25045637701771943560010771618, −2.10678331429427235314045644495, −1.96475311513815829271276222176, −0.987432708935437322418875491961, −0.955181287424311708228960913529, −0.77120399444964110859437206124, −0.36027022259873696096839049832,
0.36027022259873696096839049832, 0.77120399444964110859437206124, 0.955181287424311708228960913529, 0.987432708935437322418875491961, 1.96475311513815829271276222176, 2.10678331429427235314045644495, 2.25045637701771943560010771618, 2.26242807367851098693190799534, 2.91223682695815429669506877567, 3.39128726755421323402875748274, 3.40135721863220317049830975014, 3.41735063238555423679647488311, 4.12919432357728490153279108294, 4.15731785954268784274056877998, 4.27940793419390527658207975584, 4.45331343981079139193833447446, 4.85919671152643767163723918939, 5.17687241990234323073743359119, 5.28583680920974565499312676872, 5.45195373506211578202097913681, 5.69304113418518353496041083509, 5.97711329369706833411246555965, 6.01702274126565034885151224811, 6.02016552987791367905355228044, 6.50787008838141002125419353245
