| L(s) = 1 | − 8·2-s + 165·5-s + 508·7-s + 512·8-s − 1.32e3·10-s − 3.02e3·11-s − 5.03e3·13-s − 4.06e3·14-s − 4.09e3·16-s − 6.37e3·17-s + 3.01e3·19-s + 2.41e4·22-s + 7.56e4·23-s + 7.81e4·25-s + 4.03e4·26-s + 8.26e4·29-s + 1.74e5·31-s + 5.10e4·34-s + 8.38e4·35-s − 6.47e5·37-s − 2.41e4·38-s + 8.44e4·40-s + 3.08e5·41-s − 3.36e5·43-s − 6.04e5·46-s + 3.83e5·47-s + 8.23e5·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.590·5-s + 0.559·7-s + 0.353·8-s − 0.417·10-s − 0.685·11-s − 0.636·13-s − 0.395·14-s − 1/4·16-s − 0.314·17-s + 0.100·19-s + 0.484·22-s + 1.29·23-s + 25-s + 0.449·26-s + 0.629·29-s + 1.05·31-s + 0.222·34-s + 0.330·35-s − 2.10·37-s − 0.0713·38-s + 0.208·40-s + 0.698·41-s − 0.645·43-s − 0.916·46-s + 0.538·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.409341258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.409341258\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( 1 + p^{3} T + p^{6} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 33 p T - 2036 p^{2} T^{2} - 33 p^{8} T^{3} + p^{14} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 1763 T + p^{7} T^{2} )( 1 + 1255 T + p^{7} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3024 T - 10342595 T^{2} + 3024 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 5039 T - 37356996 T^{2} + 5039 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 3189 T + p^{7} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 1508 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 75600 T + 2310534553 T^{2} - 75600 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 82665 T - 10416374084 T^{2} - 82665 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 174892 T + 3074597553 T^{2} - 174892 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 323569 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 308118 T - 99817571957 T^{2} - 308118 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 336680 T - 158465188707 T^{2} + 336680 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 383196 T - 359783946047 T^{2} - 383196 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 760206 T + p^{7} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 2225664 T + 2464928756077 T^{2} - 2225664 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2244815 T + 1896451548204 T^{2} + 2244815 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 1473188 T - 3890428721979 T^{2} + 1473188 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 5006892 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 5898301 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 7028768 T + 30199670611665 T^{2} + 7028768 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 2651196 T - 20107210759211 T^{2} - 2651196 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6770901 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 16176386 T + 180877179542883 T^{2} + 16176386 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−11.86294178423367553101231724528, −11.16012619144602214346108691040, −10.56607809836380438973132556600, −10.18586857157900668119663358340, −10.04121683077560960670804518646, −9.006272271154885158694038166780, −8.785784853223215830564356780938, −8.481410257141197917208365841276, −7.61509864994735350430822832948, −7.06154772322515947533553879971, −6.89607523530378035248036911493, −5.62531752091792471290518769744, −5.57663560894673460425593932889, −4.49389301607599018444306885408, −4.48507694899730310043422621959, −2.92596122790365499405313959050, −2.79444027874867279335848791864, −1.73330536103609090636194954069, −1.21458739245951722063768984638, −0.37923359679752009997917289257,
0.37923359679752009997917289257, 1.21458739245951722063768984638, 1.73330536103609090636194954069, 2.79444027874867279335848791864, 2.92596122790365499405313959050, 4.48507694899730310043422621959, 4.49389301607599018444306885408, 5.57663560894673460425593932889, 5.62531752091792471290518769744, 6.89607523530378035248036911493, 7.06154772322515947533553879971, 7.61509864994735350430822832948, 8.481410257141197917208365841276, 8.785784853223215830564356780938, 9.006272271154885158694038166780, 10.04121683077560960670804518646, 10.18586857157900668119663358340, 10.56607809836380438973132556600, 11.16012619144602214346108691040, 11.86294178423367553101231724528
