| L(s) = 1 | + 28·4-s + 1.12e3·11-s − 240·16-s + 1.11e3·19-s + 9.27e3·29-s + 8.80e3·31-s + 1.37e4·41-s + 3.15e4·44-s + 3.20e4·49-s − 3.61e4·59-s + 7.95e4·61-s − 3.53e4·64-s + 8.49e3·71-s + 3.11e4·76-s − 4.38e4·79-s − 1.88e5·89-s + 2.86e5·101-s + 2.59e5·116-s + 6.32e5·121-s + 2.46e5·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 7/8·4-s + 2.81·11-s − 0.234·16-s + 0.706·19-s + 2.04·29-s + 1.64·31-s + 1.27·41-s + 2.45·44-s + 1.90·49-s − 1.35·59-s + 2.73·61-s − 1.08·64-s + 0.200·71-s + 0.618·76-s − 0.790·79-s − 2.51·89-s + 2.79·101-s + 1.79·116-s + 3.92·121-s + 1.43·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.243559876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.243559876\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 7 p^{2} T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 32014 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 564 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 335542 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2061790 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 556 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 12167086 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4638 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4400 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 132879814 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6870 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 201010150 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 110046430 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 302671514 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 18084 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 39758 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2168117590 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4248 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2456111086 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 21920 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1079748982 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 94086 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 14730169150 T^{2} + p^{10} T^{4} \) |
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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
−11.58573683061253671668194385879, −11.58477107233180282342429198677, −10.57519340272545575083735488927, −10.29600435511974412487539465361, −9.479369679675204633867685281197, −9.363298172017162168850731316273, −8.574970203066506021245938784990, −8.357590982297709008994160238173, −7.38565576248644425606633828366, −6.98460837988539779775162726257, −6.55410805989114019527245959491, −6.20383953426909369391885989058, −5.59410311062137401825533661940, −4.43623914218678824915984714842, −4.35405154574435040877517251617, −3.47189742641004006121791721538, −2.77774070830453835613338509117, −2.08581958914712638314543103455, −1.06096480391596488669266959070, −0.952989811149372150978050008911,
0.952989811149372150978050008911, 1.06096480391596488669266959070, 2.08581958914712638314543103455, 2.77774070830453835613338509117, 3.47189742641004006121791721538, 4.35405154574435040877517251617, 4.43623914218678824915984714842, 5.59410311062137401825533661940, 6.20383953426909369391885989058, 6.55410805989114019527245959491, 6.98460837988539779775162726257, 7.38565576248644425606633828366, 8.357590982297709008994160238173, 8.574970203066506021245938784990, 9.363298172017162168850731316273, 9.479369679675204633867685281197, 10.29600435511974412487539465361, 10.57519340272545575083735488927, 11.58477107233180282342429198677, 11.58573683061253671668194385879
