| L(s) = 1 | − 12·3-s + 6·5-s − 6·7-s + 90·9-s − 18·11-s − 14·13-s − 72·15-s + 8·19-s + 72·21-s + 30·23-s + 3·25-s − 540·27-s + 6·29-s + 74·31-s + 216·33-s − 36·35-s − 120·37-s + 168·39-s − 138·41-s + 10·43-s + 540·45-s + 174·47-s + 11·49-s − 108·55-s − 96·57-s − 18·59-s − 62·61-s + ⋯ |
| L(s) = 1 | − 4·3-s + 6/5·5-s − 6/7·7-s + 10·9-s − 1.63·11-s − 1.07·13-s − 4.79·15-s + 8/19·19-s + 24/7·21-s + 1.30·23-s + 3/25·25-s − 20·27-s + 6/29·29-s + 2.38·31-s + 6.54·33-s − 1.02·35-s − 3.24·37-s + 4.30·39-s − 3.36·41-s + 0.232·43-s + 12·45-s + 3.70·47-s + 0.224·49-s − 1.96·55-s − 1.68·57-s − 0.305·59-s − 1.01·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2151097010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2151097010\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 + p T )^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 6 T + 33 T^{2} - 126 T^{3} + 116 T^{4} - 126 p^{2} T^{5} + 33 p^{4} T^{6} - 6 p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 6 T + 25 T^{2} - 522 T^{3} - 4044 T^{4} - 522 p^{2} T^{5} + 25 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 18 T + 249 T^{2} + 2538 T^{3} + 18308 T^{4} + 2538 p^{2} T^{5} + 249 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 14 T - 95 T^{2} - 658 T^{3} + 22996 T^{4} - 658 p^{2} T^{5} - 95 p^{4} T^{6} + 14 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 516 T^{2} + 135302 T^{4} - 516 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 4 T + 18 p T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 30 T + 1401 T^{2} - 33030 T^{3} + 1091060 T^{4} - 33030 p^{2} T^{5} + 1401 p^{4} T^{6} - 30 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 6 T + 1409 T^{2} - 8382 T^{3} + 1254420 T^{4} - 8382 p^{2} T^{5} + 1409 p^{4} T^{6} - 6 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 74 T + 2281 T^{2} - 94202 T^{3} + 4022068 T^{4} - 94202 p^{2} T^{5} + 2281 p^{4} T^{6} - 74 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 60 T + 3254 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 69 T + 3268 T^{2} + 69 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 10 T - 2087 T^{2} + 15110 T^{3} + 1179268 T^{4} + 15110 p^{2} T^{5} - 2087 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 174 T + 16745 T^{2} - 1157622 T^{3} + 61675956 T^{4} - 1157622 p^{2} T^{5} + 16745 p^{4} T^{6} - 174 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 996 T^{2} - 9136858 T^{4} - 996 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 18 T + 6969 T^{2} + 123498 T^{3} + 35331908 T^{4} + 123498 p^{2} T^{5} + 6969 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 62 T - 4463 T^{2} + 53630 T^{3} + 40856884 T^{4} + 53630 p^{2} T^{5} - 4463 p^{4} T^{6} + 62 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 22 T + 985 T^{2} - 208538 T^{3} - 22072796 T^{4} - 208538 p^{2} T^{5} + 985 p^{4} T^{6} + 22 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 16452 T^{2} + 117605702 T^{4} - 16452 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 20 T + 7302 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 86 T - 2231 T^{2} - 245530 T^{3} + 7570612 T^{4} - 245530 p^{2} T^{5} - 2231 p^{4} T^{6} + 86 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 66 T + 9321 T^{2} + 519354 T^{3} + 24465668 T^{4} + 519354 p^{2} T^{5} + 9321 p^{4} T^{6} + 66 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 25924 T^{2} + 285535302 T^{4} - 25924 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 242 T + 25489 T^{2} - 3450194 T^{3} + 454397668 T^{4} - 3450194 p^{2} T^{5} + 25489 p^{4} T^{6} - 242 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−10.55446849568289845043810399217, −10.51955672629733634071606113790, −10.18390852892272240559590871445, −10.05830690372472960172821180596, −9.874256955944495777324945724807, −9.235921249891579579258880486514, −9.122837994284922126541428454773, −8.572971122942973807699475946149, −7.72122916025946336034105256645, −7.70966792063789311986459036415, −6.99753850693297386529697049189, −6.93641304668574602790362058676, −6.87936696258706295113441239942, −6.22772450729209262333574000741, −6.07671539573284741144079914708, −5.64066693018669042413509556559, −5.27117917312238473027574597001, −5.19515599657816200676076990932, −4.87480411787389668835497170566, −4.53777492915399931769010100391, −3.87612702068388476925627224306, −3.09948825449836774647032055976, −2.17816287193215497714178442333, −1.39329167615973852944354652640, −0.38875219257853939907942639391,
0.38875219257853939907942639391, 1.39329167615973852944354652640, 2.17816287193215497714178442333, 3.09948825449836774647032055976, 3.87612702068388476925627224306, 4.53777492915399931769010100391, 4.87480411787389668835497170566, 5.19515599657816200676076990932, 5.27117917312238473027574597001, 5.64066693018669042413509556559, 6.07671539573284741144079914708, 6.22772450729209262333574000741, 6.87936696258706295113441239942, 6.93641304668574602790362058676, 6.99753850693297386529697049189, 7.70966792063789311986459036415, 7.72122916025946336034105256645, 8.572971122942973807699475946149, 9.122837994284922126541428454773, 9.235921249891579579258880486514, 9.874256955944495777324945724807, 10.05830690372472960172821180596, 10.18390852892272240559590871445, 10.51955672629733634071606113790, 10.55446849568289845043810399217
