| L(s) = 1 | − 3.12e4·5-s + 1.00e5·7-s + 8.29e6·11-s − 1.24e7·13-s − 5.92e7·17-s + 6.90e7·19-s − 3.08e8·23-s + 7.32e8·25-s − 6.33e9·29-s + 1.23e9·31-s − 3.14e9·35-s + 1.25e10·37-s + 5.83e10·41-s − 5.28e10·43-s + 8.11e10·47-s − 8.08e10·49-s + 1.74e11·53-s − 2.59e11·55-s + 2.98e11·59-s − 3.90e11·61-s + 3.88e11·65-s − 4.38e11·67-s + 6.49e11·71-s + 3.39e11·73-s + 8.33e11·77-s − 5.47e12·79-s + 4.26e12·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.322·7-s + 1.41·11-s − 0.713·13-s − 0.595·17-s + 0.336·19-s − 0.434·23-s + 3/5·25-s − 1.97·29-s + 0.250·31-s − 0.288·35-s + 0.801·37-s + 1.91·41-s − 1.27·43-s + 1.09·47-s − 0.834·49-s + 1.08·53-s − 1.26·55-s + 0.922·59-s − 0.970·61-s + 0.638·65-s − 0.591·67-s + 0.602·71-s + 0.262·73-s + 0.455·77-s − 2.53·79-s + 1.43·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p^{6} T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 100528 T + 12988568130 p T^{2} - 100528 p^{13} T^{3} + p^{26} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 8293128 T + 5962186602578 p T^{2} - 8293128 p^{13} T^{3} + p^{26} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 12420020 T + 199064867424606 T^{2} + 12420020 p^{13} T^{3} + p^{26} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 59224788 T + 20390024845520710 T^{2} + 59224788 p^{13} T^{3} + p^{26} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 69043480 T + 45955465455476118 T^{2} - 69043480 p^{13} T^{3} + p^{26} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 308501328 T + 906428806985054062 T^{2} + 308501328 p^{13} T^{3} + p^{26} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6338832348 T + 30326805181394732254 T^{2} + 6338832348 p^{13} T^{3} + p^{26} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 1235564368 T + 16709949008135416638 T^{2} - 1235564368 p^{13} T^{3} + p^{26} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 12505462204 T + \)\(23\!\cdots\!98\)\( T^{2} - 12505462204 p^{13} T^{3} + p^{26} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 1423551660 p T + \)\(26\!\cdots\!42\)\( T^{2} - 1423551660 p^{14} T^{3} + p^{26} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 52821350072 T + \)\(39\!\cdots\!82\)\( T^{2} + 52821350072 p^{13} T^{3} + p^{26} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 81183249408 T + \)\(12\!\cdots\!70\)\( T^{2} - 81183249408 p^{13} T^{3} + p^{26} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 174532259604 T + \)\(42\!\cdots\!50\)\( T^{2} - 174532259604 p^{13} T^{3} + p^{26} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 298930858632 T + \)\(83\!\cdots\!14\)\( T^{2} - 298930858632 p^{13} T^{3} + p^{26} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 390666702596 T + \)\(33\!\cdots\!66\)\( T^{2} + 390666702596 p^{13} T^{3} + p^{26} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 438234972008 T + \)\(11\!\cdots\!90\)\( T^{2} + 438234972008 p^{13} T^{3} + p^{26} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 649973456400 T + \)\(22\!\cdots\!22\)\( T^{2} - 649973456400 p^{13} T^{3} + p^{26} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 339800917684 T + \)\(12\!\cdots\!30\)\( T^{2} - 339800917684 p^{13} T^{3} + p^{26} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5476093581584 T + \)\(15\!\cdots\!42\)\( T^{2} + 5476093581584 p^{13} T^{3} + p^{26} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4260452641512 T + \)\(19\!\cdots\!62\)\( T^{2} - 4260452641512 p^{13} T^{3} + p^{26} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1526954209260 T + \)\(20\!\cdots\!38\)\( T^{2} - 1526954209260 p^{13} T^{3} + p^{26} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 7704040415804 T + \)\(12\!\cdots\!58\)\( T^{2} + 7704040415804 p^{13} T^{3} + p^{26} T^{4} \) |
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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
−9.805068319928242213706690660113, −9.693631457204707866998599587261, −8.870552658616260062425983222442, −8.824306696615239782984171685536, −7.86806080551228032208787313867, −7.67333969285462808253368294608, −7.06797297289814296980920111946, −6.69078480330801194699059095751, −5.91663286258477767489227927934, −5.57571526499903560712340365113, −4.59632882348738477303595618625, −4.47963245337868359950186970985, −3.70016449315602839051440007742, −3.54726530613155915064700393842, −2.50088480085319733587251181997, −2.20747844919344545664915403820, −1.27878644723122172462676720914, −1.04915048855780290588695361158, 0, 0,
1.04915048855780290588695361158, 1.27878644723122172462676720914, 2.20747844919344545664915403820, 2.50088480085319733587251181997, 3.54726530613155915064700393842, 3.70016449315602839051440007742, 4.47963245337868359950186970985, 4.59632882348738477303595618625, 5.57571526499903560712340365113, 5.91663286258477767489227927934, 6.69078480330801194699059095751, 7.06797297289814296980920111946, 7.67333969285462808253368294608, 7.86806080551228032208787313867, 8.824306696615239782984171685536, 8.870552658616260062425983222442, 9.693631457204707866998599587261, 9.805068319928242213706690660113
