| L(s) = 1 | − 96·7-s + 256·13-s − 1.15e3·19-s + 384·25-s − 2.78e3·31-s + 3.25e3·37-s − 8.64e3·43-s − 260·49-s − 312·61-s − 2.36e4·67-s − 1.61e4·73-s − 1.33e4·79-s − 2.45e4·91-s − 1.12e4·97-s + 1.16e4·103-s + 4.12e4·109-s − 1.23e4·121-s + ⋯ |
| L(s) = 1 | − 1.95·7-s + 1.51·13-s − 3.19·19-s + 0.614·25-s − 2.89·31-s + 2.37·37-s − 4.67·43-s − 0.108·49-s − 0.0838·61-s − 5.26·67-s − 3.02·73-s − 2.13·79-s − 2.96·91-s − 1.19·97-s + 1.09·103-s + 3.46·109-s − 0.843·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.3128585890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3128585890\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 384 T^{2} + 237506 T^{4} - 384 p^{8} T^{6} + p^{16} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 48 T + 3586 T^{2} + 48 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 + 12348 T^{2} + 53959238 T^{4} + 12348 p^{8} T^{6} + p^{16} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 128 T + 45090 T^{2} - 128 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 265216 T^{2} + 31255382274 T^{4} - 265216 p^{8} T^{6} + p^{16} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 576 T + 314914 T^{2} + 576 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 776068 T^{2} + 289207442310 T^{4} - 776068 p^{8} T^{6} + p^{16} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2513280 T^{2} + 2575701102914 T^{4} - 2513280 p^{8} T^{6} + p^{16} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 1392 T + 1684546 T^{2} + 1392 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 44 p T + 3378726 T^{2} - 44 p^{5} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 10086912 T^{2} + 41102518325378 T^{4} - 10086912 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 4320 T + 10635874 T^{2} + 4320 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 11276292 T^{2} + 74962747331846 T^{4} - 11276292 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 25153920 T^{2} + 282144339893954 T^{4} - 25153920 p^{8} T^{6} + p^{16} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 16703940 T^{2} + 136346193182534 T^{4} - 16703940 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 156 T + 1892966 T^{2} + 156 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 11808 T + 74808226 T^{2} + 11808 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 74615940 T^{2} + 2507552925381254 T^{4} - 74615940 p^{8} T^{6} + p^{16} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 8064 T + 72795458 T^{2} + 8064 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 6672 T + 57330370 T^{2} + 6672 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 124074820 T^{2} + 7474823548026054 T^{4} - 124074820 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 118087680 T^{2} + 9299246444069762 T^{4} - 118087680 p^{8} T^{6} + p^{16} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 5632 T + 144668418 T^{2} + 5632 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
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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
−8.073877633713919946499738069209, −7.55726223657653197785083220457, −7.43595258277356786049889791505, −6.89306152559858392542893465130, −6.81797569495278978411714199187, −6.66810752737156614651796064858, −6.28894272124356691538362420318, −6.13140981297280531776239048847, −5.87816549296604209689613034331, −5.64054053765741067452465652761, −5.47809005422233041877100945391, −4.69206651439198066186893170873, −4.39729146714329278382113352630, −4.37098399955041882499460718768, −4.24468460702008277668500015937, −3.39522454069940436852808474547, −3.28995350312308167033536763595, −3.19683662103930035556860087816, −2.99186830242328832573932055501, −2.22761459475521873195880681475, −1.77049203397251060853376620652, −1.71057922227920376218280727320, −1.25620240667489009176467737673, −0.31512452330704824216506211479, −0.17375106870212800721623618225,
0.17375106870212800721623618225, 0.31512452330704824216506211479, 1.25620240667489009176467737673, 1.71057922227920376218280727320, 1.77049203397251060853376620652, 2.22761459475521873195880681475, 2.99186830242328832573932055501, 3.19683662103930035556860087816, 3.28995350312308167033536763595, 3.39522454069940436852808474547, 4.24468460702008277668500015937, 4.37098399955041882499460718768, 4.39729146714329278382113352630, 4.69206651439198066186893170873, 5.47809005422233041877100945391, 5.64054053765741067452465652761, 5.87816549296604209689613034331, 6.13140981297280531776239048847, 6.28894272124356691538362420318, 6.66810752737156614651796064858, 6.81797569495278978411714199187, 6.89306152559858392542893465130, 7.43595258277356786049889791505, 7.55726223657653197785083220457, 8.073877633713919946499738069209
