| L(s) = 1 | − 7·3-s − 32·4-s − 58·7-s + 27·9-s + 224·12-s + 640·16-s − 238·19-s + 406·21-s − 191·25-s − 224·27-s + 1.85e3·28-s − 864·36-s − 4.48e3·48-s + 1.52e3·49-s + 1.66e3·57-s − 1.56e3·63-s − 1.02e4·64-s + 1.33e3·75-s + 7.61e3·76-s + 1.67e3·79-s + 1.56e3·81-s − 1.29e4·84-s + 6.11e3·100-s + 7.16e3·108-s − 3.71e4·112-s − 5.32e3·121-s + 127-s + ⋯ |
| L(s) = 1 | − 1.34·3-s − 4·4-s − 3.13·7-s + 9-s + 5.38·12-s + 10·16-s − 2.87·19-s + 4.21·21-s − 1.52·25-s − 1.59·27-s + 12.5·28-s − 4·36-s − 13.4·48-s + 4.45·49-s + 3.87·57-s − 3.13·63-s − 20·64-s + 2.05·75-s + 11.4·76-s + 2.37·79-s + 2.15·81-s − 16.8·84-s + 6.11·100-s + 6.38·108-s − 31.3·112-s − 4·121-s + 0.000698·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 59^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 59^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0004170685410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0004170685410\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 + 7 T + 22 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 21 T + 316 T^{2} - 21 p^{3} T^{3} + p^{6} T^{4} )( 1 + 21 T + 316 T^{2} + 21 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 + 29 T + 498 T^{2} + 29 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - 126 T + p^{3} T^{2} )^{2}( 1 + 126 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 119 T + 7302 T^{2} + 119 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 29 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 159 T + 892 T^{2} - 159 p^{3} T^{3} + p^{6} T^{4} )( 1 + 159 T + 892 T^{2} + 159 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 31 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 41 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 525 T + 206704 T^{2} - 525 p^{3} T^{3} + p^{6} T^{4} )( 1 + 525 T + 206704 T^{2} + 525 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 43 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 327 T - 41948 T^{2} - 327 p^{3} T^{3} + p^{6} T^{4} )( 1 + 327 T - 41948 T^{2} + 327 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 - 180 T + p^{3} T^{2} )^{2}( 1 + 180 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 835 T + 204186 T^{2} - 835 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.850156423830596864415932079742, −8.703130671531721083388515498000, −8.567013926483412586395556568972, −7.941712549700444723198189741399, −7.88568581792547979547884420356, −7.59428558578787852750538549269, −7.16064898845484828680573886858, −6.40171858580779191946942758084, −6.38726489432904388152097064027, −6.29291062911492051506644527401, −5.98382194852621079607289776790, −5.69731312289455907248978754614, −5.08288369691708611030722406455, −5.04493462268975431818553505039, −4.96808870443704163107983524371, −4.03510180718320369584739869492, −3.94397026756928190544901243350, −3.93232725477115576020955822793, −3.83200236561842820092989515820, −3.10488000042362624741191071477, −2.70309303190026504760401059898, −1.76306720346853333465005933389, −0.914810407908609963011206365389, −0.34200826286994347660510959214, −0.01938114876531591030430636149,
0.01938114876531591030430636149, 0.34200826286994347660510959214, 0.914810407908609963011206365389, 1.76306720346853333465005933389, 2.70309303190026504760401059898, 3.10488000042362624741191071477, 3.83200236561842820092989515820, 3.93232725477115576020955822793, 3.94397026756928190544901243350, 4.03510180718320369584739869492, 4.96808870443704163107983524371, 5.04493462268975431818553505039, 5.08288369691708611030722406455, 5.69731312289455907248978754614, 5.98382194852621079607289776790, 6.29291062911492051506644527401, 6.38726489432904388152097064027, 6.40171858580779191946942758084, 7.16064898845484828680573886858, 7.59428558578787852750538549269, 7.88568581792547979547884420356, 7.941712549700444723198189741399, 8.567013926483412586395556568972, 8.703130671531721083388515498000, 8.850156423830596864415932079742
