| L(s) = 1 | − 2-s + 8·4-s − 5·5-s + 6·7-s − 23·8-s + 5·10-s + 47·11-s + 5·13-s − 6·14-s + 23·16-s − 262·17-s − 112·19-s − 40·20-s − 47·22-s − 3·23-s − 5·26-s + 48·28-s + 157·29-s − 225·31-s − 184·32-s + 262·34-s − 30·35-s − 140·37-s + 112·38-s + 115·40-s − 140·41-s − 397·43-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 4-s − 0.447·5-s + 0.323·7-s − 1.01·8-s + 0.158·10-s + 1.28·11-s + 0.106·13-s − 0.114·14-s + 0.359·16-s − 3.73·17-s − 1.35·19-s − 0.447·20-s − 0.455·22-s − 0.0271·23-s − 0.0377·26-s + 0.323·28-s + 1.00·29-s − 1.30·31-s − 1.01·32-s + 1.32·34-s − 0.144·35-s − 0.622·37-s + 0.478·38-s + 0.454·40-s − 0.533·41-s − 1.40·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5958135895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5958135895\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T - 7 T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T - 307 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 47 T + 878 T^{2} - 47 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 5 T - 2172 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 131 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 56 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 12158 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 157 T + 260 T^{2} - 157 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 225 T + 20834 T^{2} + 225 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 70 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 140 T - 49321 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 397 T + 78102 T^{2} + 397 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 347 T + 16586 T^{2} - 347 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 748 T + 354125 T^{2} + 748 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 338 T - 112737 T^{2} - 338 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 492 T - 58699 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 32 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 970 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 1257 T + 1087010 T^{2} - 1257 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 102 T - 561383 T^{2} - 102 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1488 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 974 T + 36003 T^{2} + 974 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.94308716979438142381523081130, −10.92244342972940366868759375716, −10.40352977389471248843630096273, −9.340219338216906302378144865770, −9.299338505080798643872617590484, −8.606373771847465909485087995233, −8.604770110151352397062809517034, −7.941206292556602256481770257707, −6.97229033533733779407469331485, −6.76748556731775105490301061839, −6.65491115174719910200172102357, −6.11274720338049867790055534378, −5.28601063054817385014137866245, −4.48945727991561169321278616129, −4.17242442174866633496967974189, −3.58997144572511876176638193493, −2.57493960609272390675170464096, −2.15036841508936084463792814075, −1.59904946002447965331617038683, −0.25420302159664218699094324509,
0.25420302159664218699094324509, 1.59904946002447965331617038683, 2.15036841508936084463792814075, 2.57493960609272390675170464096, 3.58997144572511876176638193493, 4.17242442174866633496967974189, 4.48945727991561169321278616129, 5.28601063054817385014137866245, 6.11274720338049867790055534378, 6.65491115174719910200172102357, 6.76748556731775105490301061839, 6.97229033533733779407469331485, 7.941206292556602256481770257707, 8.604770110151352397062809517034, 8.606373771847465909485087995233, 9.299338505080798643872617590484, 9.340219338216906302378144865770, 10.40352977389471248843630096273, 10.92244342972940366868759375716, 10.94308716979438142381523081130
