| L(s) = 1 | − 16·2-s + 170·3-s + 544·5-s − 2.72e3·6-s + 4.09e3·8-s + 1.96e4·9-s − 8.70e3·10-s − 4.88e4·11-s + 3.17e4·13-s + 9.24e4·15-s − 6.55e4·16-s − 2.14e4·17-s − 3.14e5·18-s − 7.16e5·19-s + 7.81e5·22-s + 2.47e6·23-s + 6.96e5·24-s + 1.95e6·25-s − 5.08e5·26-s + 5.12e6·27-s + 1.11e7·29-s − 1.47e6·30-s + 5.79e6·31-s − 8.30e6·33-s + 3.42e5·34-s + 3.89e6·37-s + 1.14e7·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.21·3-s + 0.389·5-s − 0.856·6-s + 0.353·8-s + 9-s − 0.275·10-s − 1.00·11-s + 0.308·13-s + 0.471·15-s − 1/4·16-s − 0.0621·17-s − 0.707·18-s − 1.26·19-s + 0.710·22-s + 1.84·23-s + 0.428·24-s + 25-s − 0.218·26-s + 1.85·27-s + 2.91·29-s − 0.333·30-s + 1.12·31-s − 1.21·33-s + 0.0439·34-s + 0.341·37-s + 0.891·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.758119953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.758119953\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_2$ | \( 1 + p^{4} T + p^{8} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 170 T + 9217 T^{2} - 170 p^{9} T^{3} + p^{18} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 544 T - 1657189 T^{2} - 544 p^{9} T^{3} + p^{18} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 48824 T + 25835285 T^{2} + 48824 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 15876 T + p^{9} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 21418 T - 118129145773 T^{2} + 21418 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 716410 T + 190555590321 T^{2} + 716410 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2470000 T + 4299747338537 T^{2} - 2470000 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5556826 T + p^{9} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 5799348 T + 7192815064433 T^{2} - 5799348 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 3894430 T - 114795154770177 T^{2} - 3894430 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6360858 T + p^{9} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 18701296 T + p^{9} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 56539068 T + 2077535737205857 T^{2} - 56539068 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 59894682 T + 287609340078991 T^{2} - 59894682 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 165629662 T + 18770189115579305 T^{2} - 165629662 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 51419016 T - 9050230886425885 T^{2} - 51419016 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 93546508 T - 18455585237300883 T^{2} + 93546508 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 95633536 T + p^{9} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 306496402 T + 35068457730677691 T^{2} - 306496402 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 496474152 T + 126634987621500785 T^{2} + 496474152 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 371486962 T + p^{9} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 165482550 T - 322971929352982709 T^{2} + 165482550 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 758016742 T + p^{9} T^{2} )^{2} \) |
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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
−12.45330353235755547236486059813, −11.99002157214376322502750534899, −10.81244273517452824280705794133, −10.71481230770850159051924591987, −10.10568917230731210793538943346, −9.724091350715035346418469053200, −8.756149618603340414563188519655, −8.533481962835343210836189350695, −8.444940429818283912121152798605, −7.54284452902635867049470989945, −6.70979904824197983516457868873, −6.62064455681018898000585962434, −5.36824884895012053231788120376, −4.70331614376073596831723991258, −4.24235621477520368486497675920, −2.90646386792010424596404935256, −2.86242646544434583849558682444, −2.09979459345909241255770644995, −0.883514697767720105872092082395, −0.853950227704393326447897344025,
0.853950227704393326447897344025, 0.883514697767720105872092082395, 2.09979459345909241255770644995, 2.86242646544434583849558682444, 2.90646386792010424596404935256, 4.24235621477520368486497675920, 4.70331614376073596831723991258, 5.36824884895012053231788120376, 6.62064455681018898000585962434, 6.70979904824197983516457868873, 7.54284452902635867049470989945, 8.444940429818283912121152798605, 8.533481962835343210836189350695, 8.756149618603340414563188519655, 9.724091350715035346418469053200, 10.10568917230731210793538943346, 10.71481230770850159051924591987, 10.81244273517452824280705794133, 11.99002157214376322502750534899, 12.45330353235755547236486059813
