| L(s) = 1 | + 54·3-s + 220·5-s − 686·7-s + 2.18e3·9-s − 2.26e3·11-s − 7.70e3·13-s + 1.18e4·15-s − 1.99e4·17-s − 1.21e4·19-s − 3.70e4·21-s − 6.57e3·23-s − 1.11e5·25-s + 7.87e4·27-s − 9.34e4·29-s − 1.45e5·31-s − 1.22e5·33-s − 1.50e5·35-s − 2.30e5·37-s − 4.15e5·39-s + 4.28e5·41-s − 4.32e5·43-s + 4.81e5·45-s − 1.67e5·47-s + 3.52e5·49-s − 1.07e6·51-s + 4.68e4·53-s − 4.98e5·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.787·5-s − 0.755·7-s + 9-s − 0.512·11-s − 0.972·13-s + 0.908·15-s − 0.983·17-s − 0.404·19-s − 0.872·21-s − 0.112·23-s − 1.42·25-s + 0.769·27-s − 0.711·29-s − 0.879·31-s − 0.592·33-s − 0.594·35-s − 0.747·37-s − 1.12·39-s + 0.970·41-s − 0.829·43-s + 0.787·45-s − 0.235·47-s + 3/7·49-s − 1.13·51-s + 0.0431·53-s − 0.403·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 44 p T + 31942 p T^{2} - 44 p^{8} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2264 T + 40039766 T^{2} + 2264 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 7700 T + 90414894 T^{2} + 7700 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 19932 T + 643164262 T^{2} + 19932 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 12104 T + 74770182 T^{2} + 12104 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6576 T + 5792085838 T^{2} + 6576 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 93492 T + 14066077774 T^{2} + 93492 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 145952 T + 8995912638 T^{2} + 145952 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 230308 T + 174630722142 T^{2} + 230308 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 428404 T + 286086237206 T^{2} - 428404 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 432728 T + 575487366870 T^{2} + 432728 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 167392 T + 998298438302 T^{2} + 167392 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 46812 T + 1516794184510 T^{2} - 46812 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1556760 T + 1205289530038 T^{2} + 1556760 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2521716 T + 6904824469646 T^{2} + 2521716 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3417416 T + 14141142792870 T^{2} + 3417416 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1455184 T - 1126426487314 T^{2} + 1455184 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2262028 T + 18853369977750 T^{2} + 2262028 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 255040 T + 11778101974878 T^{2} + 255040 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2321096 T + 55438370129798 T^{2} + 2321096 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 16345452 T + 154863742056694 T^{2} + 16345452 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16580988 T + 227654357777222 T^{2} + 16580988 p^{7} T^{3} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.93308772302339208700143959581, −10.75180774794759883808547754913, −9.839309266355383919886588379689, −9.837336951672974797218232886483, −9.117255220172037246747537986330, −8.987182930728308633945351152431, −7.922357782783084239010710409927, −7.88431135650149992311546973429, −6.86176866339559424141078604228, −6.81344927529661064075111026158, −5.61861446540322333382479926011, −5.61705295625438223950093182590, −4.30503709672150977509392787205, −4.18454795386251471987516885244, −3.00914016488186743279737373189, −2.81710802341273201899484153118, −1.84369225905458013470528894967, −1.71878134344638417579015873167, 0, 0,
1.71878134344638417579015873167, 1.84369225905458013470528894967, 2.81710802341273201899484153118, 3.00914016488186743279737373189, 4.18454795386251471987516885244, 4.30503709672150977509392787205, 5.61705295625438223950093182590, 5.61861446540322333382479926011, 6.81344927529661064075111026158, 6.86176866339559424141078604228, 7.88431135650149992311546973429, 7.922357782783084239010710409927, 8.987182930728308633945351152431, 9.117255220172037246747537986330, 9.837336951672974797218232886483, 9.839309266355383919886588379689, 10.75180774794759883808547754913, 10.93308772302339208700143959581
