| L(s) = 1 | + 16·7-s − 8·13-s + 32·19-s + 8·31-s + 136·37-s − 80·43-s + 144·49-s − 40·61-s + 304·67-s − 152·73-s + 200·79-s − 128·91-s + 424·97-s + 112·103-s + 104·109-s + 88·121-s + 127-s + 131-s + 512·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 456·169-s + ⋯ |
| L(s) = 1 | + 16/7·7-s − 0.615·13-s + 1.68·19-s + 8/31·31-s + 3.67·37-s − 1.86·43-s + 2.93·49-s − 0.655·61-s + 4.53·67-s − 2.08·73-s + 2.53·79-s − 1.40·91-s + 4.37·97-s + 1.08·103-s + 0.954·109-s + 8/11·121-s + 0.00787·127-s + 0.00763·131-s + 3.84·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(10.33277946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.33277946\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( ( 1 - 8 T + 24 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 8 p T^{2} + 18258 T^{4} - 8 p^{5} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 4 T + 252 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 290 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 16 T + 426 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1612 T^{2} + 1157478 T^{4} - 1612 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 700 T^{2} + 707622 T^{4} - 700 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 1566 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 68 T + 3084 T^{2} - 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 640 T^{2} + 3887682 T^{4} - 640 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 40 T + 3738 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 340 T^{2} - 372378 T^{4} - 340 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1876 T^{2} - 4075194 T^{4} - 1876 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 280 T^{2} - 9454638 T^{4} - 280 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 20 T + 4302 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 76 T + p^{2} T^{2} )^{4} \) |
| 71 | $D_4\times C_2$ | \( 1 - 13540 T^{2} + 89191302 T^{4} - 13540 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 76 T + 8862 T^{2} + 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 100 T + 11742 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 21652 T^{2} + 211289478 T^{4} - 21652 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 14368 T^{2} + 154232898 T^{4} - 14368 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 212 T + 29694 T^{2} - 212 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−7.26405417708852995745309141265, −6.76808251549694223700840580058, −6.51521351016166261407428660521, −6.44598469963073591769778952674, −6.17509607275490373459872795383, −5.80148675846979110668595642186, −5.48599484145732668313494919309, −5.37499993809252618084559194496, −5.32004300506729857362613978319, −4.74897906397951657671489394278, −4.68903486359397880568963450554, −4.50880242213344184829396362296, −4.49948251955235582546268017701, −3.96889367945738854238791379364, −3.52400212350291287986051095093, −3.38646489014816889115858795852, −3.23517432940308253474361644353, −2.71836241076923793614248730722, −2.28524143360053878383644318692, −2.16860864310070929605383449485, −2.01204590855192238554936650285, −1.45743337314792152466247464701, −1.04717930802465187988422061520, −0.815950685480067620169611315923, −0.54154934438378316959665716354,
0.54154934438378316959665716354, 0.815950685480067620169611315923, 1.04717930802465187988422061520, 1.45743337314792152466247464701, 2.01204590855192238554936650285, 2.16860864310070929605383449485, 2.28524143360053878383644318692, 2.71836241076923793614248730722, 3.23517432940308253474361644353, 3.38646489014816889115858795852, 3.52400212350291287986051095093, 3.96889367945738854238791379364, 4.49948251955235582546268017701, 4.50880242213344184829396362296, 4.68903486359397880568963450554, 4.74897906397951657671489394278, 5.32004300506729857362613978319, 5.37499993809252618084559194496, 5.48599484145732668313494919309, 5.80148675846979110668595642186, 6.17509607275490373459872795383, 6.44598469963073591769778952674, 6.51521351016166261407428660521, 6.76808251549694223700840580058, 7.26405417708852995745309141265
