| L(s) = 1 | − 128·4-s − 7.06e3·7-s − 8.36e4·13-s + 1.63e4·16-s − 7.26e4·19-s + 7.26e5·25-s + 9.04e5·28-s − 9.42e5·31-s − 6.01e6·37-s + 7.24e6·43-s + 2.58e7·49-s + 1.07e7·52-s − 1.08e7·61-s − 2.09e6·64-s − 1.22e7·67-s − 9.80e7·73-s + 9.29e6·76-s + 1.67e7·79-s + 5.90e8·91-s + 4.08e7·97-s − 9.30e7·100-s − 5.96e7·103-s − 9.77e7·109-s − 1.15e8·112-s + 2.15e7·121-s + 1.20e8·124-s + 127-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 2.94·7-s − 2.92·13-s + 1/4·16-s − 0.557·19-s + 1.86·25-s + 1.47·28-s − 1.02·31-s − 3.20·37-s + 2.11·43-s + 4.49·49-s + 1.46·52-s − 0.785·61-s − 1/8·64-s − 0.607·67-s − 3.45·73-s + 0.278·76-s + 0.429·79-s + 8.61·91-s + 0.461·97-s − 0.930·100-s − 0.529·103-s − 0.692·109-s − 0.735·112-s + 0.100·121-s + 0.510·124-s + 1.63·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.03803402313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03803402313\) |
| \(L(5)\) |
|
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 | $C_2$ | \( 1 + p^{7} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 29072 p^{2} T^{2} + p^{16} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3532 T + p^{8} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 21566114 T^{2} + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 41824 T + p^{8} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4967349824 T^{2} + p^{16} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 36304 T + p^{8} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 14462450590 T^{2} + p^{16} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 928028865104 T^{2} + p^{16} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 471196 T + p^{8} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3007402 T + p^{8} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 13027466643584 T^{2} + p^{16} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 3623720 T + p^{8} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 11446895562722 T^{2} + p^{16} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 18909552109520 T^{2} + p^{16} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 286434976404290 T^{2} + p^{16} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 5440630 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 6121576 T + p^{8} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 842318136694370 T^{2} + p^{16} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 49031152 T + p^{8} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8357756 T + p^{8} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1863467106641954 T^{2} + p^{16} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 3648102662700160 T^{2} + p^{16} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 20431328 T + p^{8} T^{2} )^{2} \) |
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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
−17.25949815641182139764398727204, −16.30200561471739735573125992095, −16.14968740601839567274559786637, −15.10236266723732315934257461573, −14.57023672330232238167464165491, −13.73505940951187823737649356337, −12.84130577236346837842314280934, −12.48095525625212537133604321508, −12.24558213824575401240091870920, −10.42988872452461900709699907837, −10.17015811092955557983789660745, −9.280584036770352313080083503908, −8.999896508191296451017143115245, −7.15005099945856710634261426168, −7.02064824907503988817208373729, −5.80019996508245135146267048644, −4.70653189977563922506464077158, −3.35859632618951633508319508725, −2.61585276472302449532125512482, −0.098357000076130454853808495150,
0.098357000076130454853808495150, 2.61585276472302449532125512482, 3.35859632618951633508319508725, 4.70653189977563922506464077158, 5.80019996508245135146267048644, 7.02064824907503988817208373729, 7.15005099945856710634261426168, 8.999896508191296451017143115245, 9.280584036770352313080083503908, 10.17015811092955557983789660745, 10.42988872452461900709699907837, 12.24558213824575401240091870920, 12.48095525625212537133604321508, 12.84130577236346837842314280934, 13.73505940951187823737649356337, 14.57023672330232238167464165491, 15.10236266723732315934257461573, 16.14968740601839567274559786637, 16.30200561471739735573125992095, 17.25949815641182139764398727204
