| L(s) = 1 | + 16·2-s + 6·3-s − 560·5-s + 96·6-s − 4.09e3·8-s + 1.96e4·9-s − 8.96e3·10-s + 5.41e4·11-s − 2.26e5·13-s − 3.36e3·15-s − 6.55e4·16-s − 6.26e3·17-s + 3.14e5·18-s − 2.57e5·19-s + 8.66e5·22-s + 2.66e5·23-s − 2.45e4·24-s + 1.95e6·25-s − 3.62e6·26-s + 3.54e5·27-s + 3.14e6·29-s − 5.37e4·30-s + 4.63e6·31-s + 3.24e5·33-s − 1.00e5·34-s + 1.19e7·37-s − 4.11e6·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0427·3-s − 0.400·5-s + 0.0302·6-s − 0.353·8-s + 9-s − 0.283·10-s + 1.11·11-s − 2.19·13-s − 0.0171·15-s − 1/4·16-s − 0.0181·17-s + 0.707·18-s − 0.452·19-s + 0.788·22-s + 0.198·23-s − 0.0151·24-s + 25-s − 1.55·26-s + 0.128·27-s + 0.826·29-s − 0.0121·30-s + 0.901·31-s + 0.0476·33-s − 0.0128·34-s + 1.04·37-s − 0.320·38-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
| \(L(5)\) |
\(\approx\) |
\(5.097559134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.097559134\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 p T - 2183 p^{2} T^{2} - 2 p^{10} T^{3} + p^{18} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 112 p T - 65581 p^{2} T^{2} + 112 p^{10} T^{3} + p^{18} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 54152 T + 574491413 T^{2} - 54152 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 113172 T + p^{9} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6262 T - 118548663853 T^{2} + 6262 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 257078 T - 256598599695 T^{2} + 257078 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 266000 T - 1730396661463 T^{2} - 266000 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 1574714 T + p^{9} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 4637484 T - 4933364310415 T^{2} - 4637484 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 11946238 T + 12750862557567 T^{2} - 11946238 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 21909126 T + p^{9} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 27520592 T + p^{9} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 52927836 T + 1682225350540129 T^{2} + 52927836 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 16221222 T - 3036635548628849 T^{2} + 16221222 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 140509618 T + 11079956931850985 T^{2} - 140509618 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 202963560 T + 29500060595039459 T^{2} - 202963560 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 153734572 T - 3572215768271763 T^{2} + 153734572 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3938816 p T + p^{9} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 404022830 T + 104362860452940987 T^{2} - 404022830 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 130689816 T - 102771767976504463 T^{2} - 130689816 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 420134014 T + p^{9} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 469542390 T - 129886347700573109 T^{2} - 469542390 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 872501690 T + p^{9} T^{2} )^{2} \) |
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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
−12.31648491966215198883299541082, −12.31377659562315199167610958342, −11.21032702198106835410692582258, −11.11163265956114710154072667396, −9.938857638964216529498951685198, −9.856587672622495908841259744738, −9.221423674953711280607329796132, −8.589207119582646278380693341788, −7.65330778295206546852670825937, −7.46595616542399331706859180227, −6.48427084751048912876400324667, −6.44109128350724181811848065330, −5.13887088922836023432701917768, −4.81020535206744962485018822472, −4.16723520551407997653166404845, −3.77590167184934447776544958662, −2.52847649096980083939610432545, −2.41026522082941865530767759208, −0.985137403339233008378496213823, −0.67138322272956082428821931454,
0.67138322272956082428821931454, 0.985137403339233008378496213823, 2.41026522082941865530767759208, 2.52847649096980083939610432545, 3.77590167184934447776544958662, 4.16723520551407997653166404845, 4.81020535206744962485018822472, 5.13887088922836023432701917768, 6.44109128350724181811848065330, 6.48427084751048912876400324667, 7.46595616542399331706859180227, 7.65330778295206546852670825937, 8.589207119582646278380693341788, 9.221423674953711280607329796132, 9.856587672622495908841259744738, 9.938857638964216529498951685198, 11.11163265956114710154072667396, 11.21032702198106835410692582258, 12.31377659562315199167610958342, 12.31648491966215198883299541082
