| L(s) = 1 | − 16·4-s + 486·9-s − 500·11-s + 256·16-s + 156·19-s − 5.15e3·29-s − 1.73e4·31-s − 7.77e3·36-s + 2.10e3·41-s + 8.00e3·44-s + 4.71e3·49-s + 2.78e4·59-s − 6.57e4·61-s − 4.09e3·64-s − 1.01e5·71-s − 2.49e3·76-s + 3.86e4·79-s + 1.77e5·81-s − 1.88e5·89-s − 2.43e5·99-s − 3.70e4·101-s + 3.57e5·109-s + 8.24e4·116-s − 1.34e5·121-s + 2.76e5·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2·9-s − 1.24·11-s + 1/4·16-s + 0.0991·19-s − 1.13·29-s − 3.23·31-s − 36-s + 0.195·41-s + 0.622·44-s + 0.280·49-s + 1.04·59-s − 2.26·61-s − 1/8·64-s − 2.37·71-s − 0.0495·76-s + 0.697·79-s + 3·81-s − 2.52·89-s − 2.49·99-s − 0.361·101-s + 2.88·109-s + 0.569·116-s − 0.835·121-s + 1.61·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 422500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3721922369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3721922369\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 4714 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 250 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 1711870 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 78 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10388910 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2578 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8654 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 17995718 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 1050 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 259206886 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 423144570 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 7279130 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 13922 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 32882 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 2139178142 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 50542 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 1960580686 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 19348 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 232677442 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 94170 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 16236041282 T^{2} + p^{10} T^{4} \) |
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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
−10.20342167150840282451778053937, −9.338332789163391110963464133790, −9.327414086266412821706191588042, −8.881311486413362419027762561393, −8.156622463232810170321130873854, −7.62842051931812832181152162594, −7.38754431193185287615937658385, −7.19458845854385260263934825074, −6.49492079448778504743548669428, −5.81876445239945086251291291849, −5.36825008280670522608614138551, −5.09010932719553827065645751240, −4.24330809710752194713377085436, −4.18269143092090988590227267580, −3.48944317235194583476508427491, −2.94276659651090590896011864796, −1.99381650084970716669695554880, −1.73768026712978313587296239269, −1.06935338525208003745537233715, −0.13907298106046146538449847430,
0.13907298106046146538449847430, 1.06935338525208003745537233715, 1.73768026712978313587296239269, 1.99381650084970716669695554880, 2.94276659651090590896011864796, 3.48944317235194583476508427491, 4.18269143092090988590227267580, 4.24330809710752194713377085436, 5.09010932719553827065645751240, 5.36825008280670522608614138551, 5.81876445239945086251291291849, 6.49492079448778504743548669428, 7.19458845854385260263934825074, 7.38754431193185287615937658385, 7.62842051931812832181152162594, 8.156622463232810170321130873854, 8.881311486413362419027762561393, 9.327414086266412821706191588042, 9.338332789163391110963464133790, 10.20342167150840282451778053937
