| L(s) = 1 | + 3·3-s + 9·5-s − 7-s − 3·9-s − 36·11-s + 5·13-s + 27·15-s + 2·19-s − 3·21-s + 99·23-s + 22·25-s − 18·27-s − 63·29-s − 7·31-s − 108·33-s − 9·35-s − 64·37-s + 15·39-s − 18·41-s − 46·43-s − 27·45-s − 81·47-s + 24·49-s − 324·55-s + 6·57-s + 126·59-s + 41·61-s + ⋯ |
| L(s) = 1 | + 3-s + 9/5·5-s − 1/7·7-s − 1/3·9-s − 3.27·11-s + 5/13·13-s + 9/5·15-s + 2/19·19-s − 1/7·21-s + 4.30·23-s + 0.879·25-s − 2/3·27-s − 2.17·29-s − 0.225·31-s − 3.27·33-s − 0.257·35-s − 1.72·37-s + 5/13·39-s − 0.439·41-s − 1.06·43-s − 3/5·45-s − 1.72·47-s + 0.489·49-s − 5.89·55-s + 2/19·57-s + 2.13·59-s + 0.672·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.560137790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.560137790\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - p T + 4 p T^{2} - p^{3} T^{3} + p^{4} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 9 T + 59 T^{2} - 288 T^{3} + 1074 T^{4} - 288 p^{2} T^{5} + 59 p^{4} T^{6} - 9 p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + T - 23 T^{2} - 74 T^{3} - 1874 T^{4} - 74 p^{2} T^{5} - 23 p^{4} T^{6} + p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 36 T + 683 T^{2} + 9036 T^{3} + 100632 T^{4} + 9036 p^{2} T^{5} + 683 p^{4} T^{6} + 36 p^{6} T^{7} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 5 T - 245 T^{2} + 340 T^{3} + 40114 T^{4} + 340 p^{2} T^{5} - 245 p^{4} T^{6} - 5 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 769 T^{2} + 298176 T^{4} - 769 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - T + 648 T^{2} - p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 99 T + 5117 T^{2} - 183150 T^{3} + 4870902 T^{4} - 183150 p^{2} T^{5} + 5117 p^{4} T^{6} - 99 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 63 T + 2123 T^{2} + 50400 T^{3} + 1045362 T^{4} + 50400 p^{2} T^{5} + 2123 p^{4} T^{6} + 63 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 7 T - 1217 T^{2} - 4592 T^{3} + 632146 T^{4} - 4592 p^{2} T^{5} - 1217 p^{4} T^{6} + 7 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 32 T + 1806 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 18 T + 1913 T^{2} + 32490 T^{3} + 613812 T^{4} + 32490 p^{2} T^{5} + 1913 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 23 T - 1320 T^{2} + 23 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 81 T + 6929 T^{2} + 384102 T^{3} + 22437966 T^{4} + 384102 p^{2} T^{5} + 6929 p^{4} T^{6} + 81 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 7204 T^{2} + 26018214 T^{4} - 7204 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 126 T + 11993 T^{2} - 844326 T^{3} + 51207492 T^{4} - 844326 p^{2} T^{5} + 11993 p^{4} T^{6} - 126 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 41 T - 5513 T^{2} + 10168 T^{3} + 31652794 T^{4} + 10168 p^{2} T^{5} - 5513 p^{4} T^{6} - 41 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 116 T + 3787 T^{2} - 80156 T^{3} + 12934456 T^{4} - 80156 p^{2} T^{5} + 3787 p^{4} T^{6} - 116 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 18616 T^{2} + 137194926 T^{4} - 18616 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 43 T + 10452 T^{2} - 43 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 83 T - 5459 T^{2} + 11122 T^{3} + 70528774 T^{4} + 11122 p^{2} T^{5} - 5459 p^{4} T^{6} - 83 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 81 T + 16289 T^{2} + 1142262 T^{3} + 166474326 T^{4} + 1142262 p^{2} T^{5} + 16289 p^{4} T^{6} + 81 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 6916 T^{2} + 69013446 T^{4} - 6916 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 196 T + 10291 T^{2} + 1824172 T^{3} + 341030200 T^{4} + 1824172 p^{2} T^{5} + 10291 p^{4} T^{6} + 196 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−12.56700517165280561925497593757, −11.63160118847508671367372263975, −11.47694210921832043824015101218, −10.93161879084923963863614374360, −10.92388431095590021738856817061, −10.39074282721924029658561171899, −10.28505753112685403611954516339, −9.625803875974266392545386611428, −9.555882657032207902577351507934, −9.248185180654348138590714608439, −8.564714299984231542017045099162, −8.480493950718427862156277348009, −8.306082582376046738474800724984, −7.54773451554962177434978772190, −7.27587624631547979538705340350, −6.90344025080803101988364865456, −6.45509526001152787510918040999, −5.54298477534478364165456106683, −5.50393391450211072285070926941, −5.08419366454016553049499615652, −5.01102340857181410439282506004, −3.37995410589960562088165736252, −3.28606872193177282185519833530, −2.50959671808882599528232416564, −2.07470438107883879783765653509,
2.07470438107883879783765653509, 2.50959671808882599528232416564, 3.28606872193177282185519833530, 3.37995410589960562088165736252, 5.01102340857181410439282506004, 5.08419366454016553049499615652, 5.50393391450211072285070926941, 5.54298477534478364165456106683, 6.45509526001152787510918040999, 6.90344025080803101988364865456, 7.27587624631547979538705340350, 7.54773451554962177434978772190, 8.306082582376046738474800724984, 8.480493950718427862156277348009, 8.564714299984231542017045099162, 9.248185180654348138590714608439, 9.555882657032207902577351507934, 9.625803875974266392545386611428, 10.28505753112685403611954516339, 10.39074282721924029658561171899, 10.92388431095590021738856817061, 10.93161879084923963863614374360, 11.47694210921832043824015101218, 11.63160118847508671367372263975, 12.56700517165280561925497593757
