| L(s) = 1 | + 540·4-s + 2.92e4·11-s + 2.94e4·16-s − 8.83e5·19-s − 2.09e6·29-s − 1.58e7·31-s − 2.65e7·41-s + 1.58e7·44-s + 4.48e7·49-s − 6.40e7·59-s + 2.21e8·61-s − 1.25e8·64-s − 5.53e8·71-s − 4.77e8·76-s − 8.96e8·79-s + 3.79e8·89-s + 2.63e9·101-s − 3.95e9·109-s − 1.13e9·116-s − 4.07e9·121-s − 8.54e9·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.05·4-s + 0.603·11-s + 0.112·16-s − 1.55·19-s − 0.551·29-s − 3.07·31-s − 1.46·41-s + 0.636·44-s + 1.11·49-s − 0.688·59-s + 2.04·61-s − 0.936·64-s − 2.58·71-s − 1.64·76-s − 2.58·79-s + 0.641·89-s + 2.51·101-s − 2.68·109-s − 0.581·116-s − 1.72·121-s − 3.24·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.08346178304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08346178304\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 135 p^{2} T^{2} + p^{18} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 44851070 T^{2} + p^{18} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 14648 T + p^{9} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 19772133910 T^{2} + p^{18} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 42608307390 T^{2} + p^{18} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 441820 T + p^{9} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 1524006503570 T^{2} + p^{18} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 1049350 T + p^{9} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7910568 T + p^{9} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 180764011793210 T^{2} + p^{18} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 13285562 T + p^{9} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 470152980649990 T^{2} + p^{18} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2045781727484190 T^{2} + p^{18} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3277713901593990 T^{2} + p^{18} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 32042120 T + p^{9} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 110664022 T + p^{9} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 40354634616070070 T^{2} + p^{18} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 276679712 T + p^{9} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 48034873641925390 T^{2} + p^{18} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 448202760 T + p^{9} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 350347374506432810 T^{2} + p^{18} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 189894930 T + p^{9} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 491963359241209150 T^{2} + p^{18} T^{4} \) |
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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
−10.84596692339171629293910255517, −10.43987571249715252952776585275, −10.12654490185219108987798355544, −9.141207662342788900242285890701, −9.041896229390973807230725744524, −8.508742896170141491667174171893, −7.76440111863353802031633541563, −7.25573203201931993515356773790, −6.88358508767655009286001259978, −6.49612687971401268647243655249, −5.70616481662662241234010215974, −5.54818740677476479596931532408, −4.56710931278375865737117193944, −4.03789626237209066435159452834, −3.51550199693124841666021900692, −2.84308566105805512912458925486, −2.02124372230029467650659929642, −1.88614505716930291086176720221, −1.19979607484612354070181669072, −0.05552853578543732343259639250,
0.05552853578543732343259639250, 1.19979607484612354070181669072, 1.88614505716930291086176720221, 2.02124372230029467650659929642, 2.84308566105805512912458925486, 3.51550199693124841666021900692, 4.03789626237209066435159452834, 4.56710931278375865737117193944, 5.54818740677476479596931532408, 5.70616481662662241234010215974, 6.49612687971401268647243655249, 6.88358508767655009286001259978, 7.25573203201931993515356773790, 7.76440111863353802031633541563, 8.508742896170141491667174171893, 9.041896229390973807230725744524, 9.141207662342788900242285890701, 10.12654490185219108987798355544, 10.43987571249715252952776585275, 10.84596692339171629293910255517
