| L(s) = 1 | + 4·2-s − 9·3-s + 86·5-s − 36·6-s − 64·8-s + 344·10-s − 34·11-s + 6·13-s − 774·15-s − 256·16-s − 1.90e3·17-s − 1.48e3·19-s − 136·22-s + 224·23-s + 576·24-s + 3.12e3·25-s + 24·26-s + 729·27-s − 1.30e4·29-s − 3.09e3·30-s + 1.73e3·31-s + 306·33-s − 7.61e3·34-s + 7.63e3·37-s − 5.95e3·38-s − 54·39-s − 5.50e3·40-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1.53·5-s − 0.408·6-s − 0.353·8-s + 1.08·10-s − 0.0847·11-s + 0.00984·13-s − 0.888·15-s − 1/4·16-s − 1.59·17-s − 0.946·19-s − 0.0599·22-s + 0.0882·23-s + 0.204·24-s + 25-s + 0.00696·26-s + 0.192·27-s − 2.87·29-s − 0.628·30-s + 0.323·31-s + 0.0489·33-s − 1.12·34-s + 0.916·37-s − 0.669·38-s − 0.00568·39-s − 0.543·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.367228021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.367228021\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 - p^{2} T + p^{4} T^{2} \) |
| 3 | $C_2$ | \( 1 + p^{2} T + p^{4} T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 86 T + 4271 T^{2} - 86 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 34 T - 159895 T^{2} + 34 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 112 p T + 7631 p^{2} T^{2} + 112 p^{6} T^{3} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 1489 T - 258978 T^{2} + 1489 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 224 T - 6386167 T^{2} - 224 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6508 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1731 T - 25632790 T^{2} - 1731 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 7633 T - 11081268 T^{2} - 7633 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 15414 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 18491 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 18462 T + 111500437 T^{2} - 18462 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 19956 T - 19953557 T^{2} - 19956 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 31828 T + 298097285 T^{2} + 31828 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 57654 T + 2479387415 T^{2} + 57654 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 60563 T + 2317751862 T^{2} - 60563 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 44834 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 20821 T - 1639557552 T^{2} - 20821 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 30531 T - 2144914438 T^{2} - 30531 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 110602 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 58992 T - 2104003385 T^{2} + 58992 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 119846 T + p^{5} T^{2} )^{2} \) |
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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
−11.27387736229650752234088444444, −10.71931116464294747510896777831, −10.35552324586099047025922338445, −9.768045403887808591763810122435, −9.306327293045087413871770529446, −8.774284993239319606535949538733, −8.682455427172272337749185291269, −7.44498123599887760610706494109, −7.29495143853881468371535156304, −6.43624624542158109095244120233, −5.93230521568187760191337984999, −5.91022029342992991163260945346, −5.23193735000298371912898166415, −4.56765105067927125306035083518, −4.19065861651640633201959575238, −3.39851478785159545803861195973, −2.41966581545510576503914260785, −2.14641973567871489123447945075, −1.40329686489076269739106779520, −0.27647657844965387612131433877,
0.27647657844965387612131433877, 1.40329686489076269739106779520, 2.14641973567871489123447945075, 2.41966581545510576503914260785, 3.39851478785159545803861195973, 4.19065861651640633201959575238, 4.56765105067927125306035083518, 5.23193735000298371912898166415, 5.91022029342992991163260945346, 5.93230521568187760191337984999, 6.43624624542158109095244120233, 7.29495143853881468371535156304, 7.44498123599887760610706494109, 8.682455427172272337749185291269, 8.774284993239319606535949538733, 9.306327293045087413871770529446, 9.768045403887808591763810122435, 10.35552324586099047025922338445, 10.71931116464294747510896777831, 11.27387736229650752234088444444
