| L(s) = 1 | + 5·4-s + 10·9-s − 12·11-s + 12·16-s + 10·19-s + 10·29-s − 12·31-s + 50·36-s − 12·41-s − 60·44-s + 25·49-s + 30·59-s + 8·61-s + 15·64-s − 22·71-s + 50·76-s + 10·79-s + 57·81-s − 120·99-s − 12·101-s − 40·109-s + 50·116-s + 56·121-s − 60·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 5/2·4-s + 10/3·9-s − 3.61·11-s + 3·16-s + 2.29·19-s + 1.85·29-s − 2.15·31-s + 25/3·36-s − 1.87·41-s − 9.04·44-s + 25/7·49-s + 3.90·59-s + 1.02·61-s + 15/8·64-s − 2.61·71-s + 5.73·76-s + 1.12·79-s + 19/3·81-s − 12.0·99-s − 1.19·101-s − 3.83·109-s + 4.64·116-s + 5.09·121-s − 5.38·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.948798358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.948798358\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 5 T^{2} + 13 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 25 T^{2} + 253 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_4$ | \( ( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 25 T^{2} + 393 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 798 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 5 T + 33 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 10 T^{2} + 363 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 130 T^{2} + 6883 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 145 T^{2} + 8853 T^{4} - 145 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 185 T^{2} + 12973 T^{4} - 185 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 170 T^{2} + 12763 T^{4} - 170 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 15 T + 163 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T + 81 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 160 T^{2} + 14398 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 11 T + 171 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 5 T + 133 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 290 T^{2} + 34483 T^{4} - 290 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 365 T^{2} + 52113 T^{4} - 365 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.49007802110720543897693615563, −7.48428064120398438121225915356, −7.24151197158025580684696081866, −6.93351606624168514841384955977, −6.77031415610353377858619929978, −6.65717171646911595805736848425, −6.55920480133898467989065774296, −5.87283574654788301851021878817, −5.59317323716215296973442642715, −5.44936856000700956702498363574, −5.22431113126404089311311913072, −5.19128497517591427854368968062, −4.85306140926171489508089612386, −4.36563928352683698584166063216, −4.07597015596526848489318655256, −3.73755592235147528671333569411, −3.67893552820163325018789607638, −2.92062907902905855595963599408, −2.91673754300495098929424786061, −2.56898862532313712441395724986, −2.25738003964280298841056092760, −2.11471709929814528298857042533, −1.45905467638233249020922176924, −1.36663032069135782685070730924, −0.72262107937754637892936497162,
0.72262107937754637892936497162, 1.36663032069135782685070730924, 1.45905467638233249020922176924, 2.11471709929814528298857042533, 2.25738003964280298841056092760, 2.56898862532313712441395724986, 2.91673754300495098929424786061, 2.92062907902905855595963599408, 3.67893552820163325018789607638, 3.73755592235147528671333569411, 4.07597015596526848489318655256, 4.36563928352683698584166063216, 4.85306140926171489508089612386, 5.19128497517591427854368968062, 5.22431113126404089311311913072, 5.44936856000700956702498363574, 5.59317323716215296973442642715, 5.87283574654788301851021878817, 6.55920480133898467989065774296, 6.65717171646911595805736848425, 6.77031415610353377858619929978, 6.93351606624168514841384955977, 7.24151197158025580684696081866, 7.48428064120398438121225915356, 7.49007802110720543897693615563
