L(s) = 1 | − 2-s − 4·3-s − 3·5-s + 4·6-s + 4·7-s + 2·8-s + 10·9-s + 3·10-s − 2·11-s − 4·14-s + 12·15-s − 16-s − 10·17-s − 10·18-s − 7·19-s − 16·21-s + 2·22-s + 3·23-s − 8·24-s + 3·25-s − 20·27-s − 9·29-s − 12·30-s + 31-s − 32-s + 8·33-s + 10·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 2.30·3-s − 1.34·5-s + 1.63·6-s + 1.51·7-s + 0.707·8-s + 10/3·9-s + 0.948·10-s − 0.603·11-s − 1.06·14-s + 3.09·15-s − 1/4·16-s − 2.42·17-s − 2.35·18-s − 1.60·19-s − 3.49·21-s + 0.426·22-s + 0.625·23-s − 1.63·24-s + 3/5·25-s − 3.84·27-s − 1.67·29-s − 2.19·30-s + 0.179·31-s − 0.176·32-s + 1.39·33-s + 1.71·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| 13 | | \( 1 \) |
good | 2 | $C_2 \wr S_4$ | \( 1 + T + T^{2} - T^{3} - p T^{4} - p T^{5} + p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 3 T + 6 T^{2} + 17 T^{3} + 62 T^{4} + 17 p T^{5} + 6 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 2 T + 24 T^{2} + 6 T^{3} + 254 T^{4} + 6 p T^{5} + 24 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 10 T + 76 T^{2} + 398 T^{3} + 1846 T^{4} + 398 p T^{5} + 76 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 7 T + 52 T^{2} + 303 T^{3} + 1286 T^{4} + 303 p T^{5} + 52 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 3 T + 14 T^{2} - 67 T^{3} + 938 T^{4} - 67 p T^{5} + 14 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 9 T + 102 T^{2} + 539 T^{3} + 3794 T^{4} + 539 p T^{5} + 102 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - T + 80 T^{2} + 19 T^{3} + 2974 T^{4} + 19 p T^{5} + 80 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 4 T + 68 T^{2} + 36 T^{3} + 1782 T^{4} + 36 p T^{5} + 68 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 28 T + 432 T^{2} + 4464 T^{3} + 33398 T^{4} + 4464 p T^{5} + 432 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 17 T + 236 T^{2} + 2033 T^{3} + 15702 T^{4} + 2033 p T^{5} + 236 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 3 T + 72 T^{2} + 367 T^{3} + 4946 T^{4} + 367 p T^{5} + 72 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 5 T + 70 T^{2} + 439 T^{3} + 2514 T^{4} + 439 p T^{5} + 70 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 28 T + 504 T^{2} + 5952 T^{3} + 53678 T^{4} + 5952 p T^{5} + 504 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 10 T + 216 T^{2} + 1694 T^{3} + 19230 T^{4} + 1694 p T^{5} + 216 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 22 T + 284 T^{2} + 2534 T^{3} + 20950 T^{4} + 2534 p T^{5} + 284 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 2 T + 264 T^{2} + 366 T^{3} + 27374 T^{4} + 366 p T^{5} + 264 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 31 T + 620 T^{2} - 8237 T^{3} + 82150 T^{4} - 8237 p T^{5} + 620 p^{2} T^{6} - 31 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + T + 284 T^{2} + 133 T^{3} + 32310 T^{4} + 133 p T^{5} + 284 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 3 T + 216 T^{2} + 691 T^{3} + 24674 T^{4} + 691 p T^{5} + 216 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 5 T + 242 T^{2} + 651 T^{3} + 26846 T^{4} + 651 p T^{5} + 242 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 43 T + 1036 T^{2} - 16593 T^{3} + 191270 T^{4} - 16593 p T^{5} + 1036 p^{2} T^{6} - 43 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.44942218049928812227011731046, −6.36032801438433117282590566458, −6.14239030828510757864899563552, −5.82306410409547164803667739961, −5.82056521182998710470578862615, −5.16673711019638889835081162210, −5.15726104353520628918801009581, −5.08553817070352006769934659725, −4.82151635523446890151150816245, −4.67382022298730165817329914179, −4.49467434349701712568704723454, −4.48125553404344436120790001634, −4.36743020913385218242698887899, −3.74154319377687297618384346316, −3.59914451597717953899000138551, −3.56974162883042052878134715662, −3.25870741364240731578751295095, −2.98904227630736429206481796971, −2.41968367367793173768516873478, −2.14024274886916768701661634990, −1.94933703246615733456041821561, −1.81227849698730758177160540804, −1.42328077418340625364709405609, −1.37646112996936646364437603292, −0.907735287282731237559980362312, 0, 0, 0, 0,
0.907735287282731237559980362312, 1.37646112996936646364437603292, 1.42328077418340625364709405609, 1.81227849698730758177160540804, 1.94933703246615733456041821561, 2.14024274886916768701661634990, 2.41968367367793173768516873478, 2.98904227630736429206481796971, 3.25870741364240731578751295095, 3.56974162883042052878134715662, 3.59914451597717953899000138551, 3.74154319377687297618384346316, 4.36743020913385218242698887899, 4.48125553404344436120790001634, 4.49467434349701712568704723454, 4.67382022298730165817329914179, 4.82151635523446890151150816245, 5.08553817070352006769934659725, 5.15726104353520628918801009581, 5.16673711019638889835081162210, 5.82056521182998710470578862615, 5.82306410409547164803667739961, 6.14239030828510757864899563552, 6.36032801438433117282590566458, 6.44942218049928812227011731046