| L(s) = 1 | + 5·3-s − 2·4-s + 3·5-s + 8-s + 15·9-s − 11-s − 10·12-s + 5·13-s + 15·15-s − 16-s + 13·17-s + 7·19-s − 6·20-s − 4·23-s + 5·24-s + 35·27-s − 12·29-s + 6·31-s + 32-s − 5·33-s − 30·36-s + 11·37-s + 25·39-s + 3·40-s + 10·41-s + 10·43-s + 2·44-s + ⋯ |
| L(s) = 1 | + 2.88·3-s − 4-s + 1.34·5-s + 0.353·8-s + 5·9-s − 0.301·11-s − 2.88·12-s + 1.38·13-s + 3.87·15-s − 1/4·16-s + 3.15·17-s + 1.60·19-s − 1.34·20-s − 0.834·23-s + 1.02·24-s + 6.73·27-s − 2.22·29-s + 1.07·31-s + 0.176·32-s − 0.870·33-s − 5·36-s + 1.80·37-s + 4.00·39-s + 0.474·40-s + 1.56·41-s + 1.52·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 7^{10} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 7^{10} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(44.77457523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(44.77457523\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{5} \) |
| 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{5} \) |
| good | 2 | $C_2 \wr S_5$ | \( 1 + p T^{2} - T^{3} + 5 T^{4} - 5 T^{5} + 5 p T^{6} - p^{2} T^{7} + p^{4} T^{8} + p^{5} T^{10} \) |
| 5 | $C_2 \wr S_5$ | \( 1 - 3 T + 9 T^{2} - 13 T^{3} + 58 T^{4} - 124 T^{5} + 58 p T^{6} - 13 p^{2} T^{7} + 9 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \) |
| 11 | $C_2 \wr S_5$ | \( 1 + T + 32 T^{2} + 82 T^{3} + 439 T^{4} + 1561 T^{5} + 439 p T^{6} + 82 p^{2} T^{7} + 32 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 - 13 T + 140 T^{2} - 966 T^{3} + 5717 T^{4} - 25329 T^{5} + 5717 p T^{6} - 966 p^{2} T^{7} + 140 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \) |
| 19 | $C_2 \wr S_5$ | \( 1 - 7 T + 89 T^{2} - 426 T^{3} + 3129 T^{4} - 587 p T^{5} + 3129 p T^{6} - 426 p^{2} T^{7} + 89 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \) |
| 23 | $C_2 \wr S_5$ | \( 1 + 4 T + 68 T^{2} + 225 T^{3} + 2455 T^{4} + 7226 T^{5} + 2455 p T^{6} + 225 p^{2} T^{7} + 68 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 + 12 T + 147 T^{2} + 1161 T^{3} + 8047 T^{4} + 46903 T^{5} + 8047 p T^{6} + 1161 p^{2} T^{7} + 147 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \) |
| 31 | $C_2 \wr S_5$ | \( 1 - 6 T + 116 T^{2} - 493 T^{3} + 6103 T^{4} - 20410 T^{5} + 6103 p T^{6} - 493 p^{2} T^{7} + 116 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \) |
| 37 | $C_2 \wr S_5$ | \( 1 - 11 T + 3 p T^{2} - 277 T^{3} - 164 T^{4} + 17076 T^{5} - 164 p T^{6} - 277 p^{2} T^{7} + 3 p^{4} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 - 10 T + 192 T^{2} - 1361 T^{3} + 14819 T^{4} - 78206 T^{5} + 14819 p T^{6} - 1361 p^{2} T^{7} + 192 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 - 10 T + 178 T^{2} - 1307 T^{3} + 303 p T^{4} - 75498 T^{5} + 303 p^{2} T^{6} - 1307 p^{2} T^{7} + 178 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 + 4 T + 64 T^{2} + 574 T^{3} + 5511 T^{4} + 17892 T^{5} + 5511 p T^{6} + 574 p^{2} T^{7} + 64 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 + 9 T + 56 T^{2} - 52 T^{3} + 3581 T^{4} + 27955 T^{5} + 3581 p T^{6} - 52 p^{2} T^{7} + 56 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 - 7 T + 70 T^{2} - 252 T^{3} + 6479 T^{4} - 52809 T^{5} + 6479 p T^{6} - 252 p^{2} T^{7} + 70 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 - 23 T + 504 T^{2} - 6416 T^{3} + 75133 T^{4} - 612637 T^{5} + 75133 p T^{6} - 6416 p^{2} T^{7} + 504 p^{3} T^{8} - 23 p^{4} T^{9} + p^{5} T^{10} \) |
| 67 | $C_2 \wr S_5$ | \( 1 - 25 T + 512 T^{2} - 6672 T^{3} + 76147 T^{4} - 658295 T^{5} + 76147 p T^{6} - 6672 p^{2} T^{7} + 512 p^{3} T^{8} - 25 p^{4} T^{9} + p^{5} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 + 27 T + 560 T^{2} + 7874 T^{3} + 92451 T^{4} + 846655 T^{5} + 92451 p T^{6} + 7874 p^{2} T^{7} + 560 p^{3} T^{8} + 27 p^{4} T^{9} + p^{5} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 - 18 T + 426 T^{2} - 4685 T^{3} + 62369 T^{4} - 484698 T^{5} + 62369 p T^{6} - 4685 p^{2} T^{7} + 426 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 - 8 T + 166 T^{2} - 1469 T^{3} + 19625 T^{4} - 113674 T^{5} + 19625 p T^{6} - 1469 p^{2} T^{7} + 166 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 - 12 T + 270 T^{2} - 1881 T^{3} + 29929 T^{4} - 164158 T^{5} + 29929 p T^{6} - 1881 p^{2} T^{7} + 270 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 - 29 T + 747 T^{2} - 11745 T^{3} + 162828 T^{4} - 1633336 T^{5} + 162828 p T^{6} - 11745 p^{2} T^{7} + 747 p^{3} T^{8} - 29 p^{4} T^{9} + p^{5} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 - 13 T + 299 T^{2} - 2005 T^{3} + 33160 T^{4} - 158644 T^{5} + 33160 p T^{6} - 2005 p^{2} T^{7} + 299 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.46867620700542721019467214194, −5.20310203439366722656185405504, −5.19576519823176809747531435691, −5.06922482594388236883415632511, −4.97818252585471738876309417596, −4.36032675400535907315894312034, −4.17941236448272417310710874317, −4.15393446546204452426839232373, −4.10315912295299274464107999607, −4.00088367559596072844068591007, −3.46329413596686342534483263780, −3.45932709248879316175025640366, −3.40603050020536800684714228040, −3.23660876834904825534270537880, −2.79464739626052682017834700317, −2.71567289116766517503981629000, −2.59959905080210460243345787380, −2.20372300896817880380840900283, −1.97685858521331521560848807983, −1.92521511251875494213025229481, −1.77730740330522352707491221366, −1.08228552817694683248334058361, −1.07055369890353248797816989338, −0.968811423314922648686546831147, −0.66543060780063271262703265358,
0.66543060780063271262703265358, 0.968811423314922648686546831147, 1.07055369890353248797816989338, 1.08228552817694683248334058361, 1.77730740330522352707491221366, 1.92521511251875494213025229481, 1.97685858521331521560848807983, 2.20372300896817880380840900283, 2.59959905080210460243345787380, 2.71567289116766517503981629000, 2.79464739626052682017834700317, 3.23660876834904825534270537880, 3.40603050020536800684714228040, 3.45932709248879316175025640366, 3.46329413596686342534483263780, 4.00088367559596072844068591007, 4.10315912295299274464107999607, 4.15393446546204452426839232373, 4.17941236448272417310710874317, 4.36032675400535907315894312034, 4.97818252585471738876309417596, 5.06922482594388236883415632511, 5.19576519823176809747531435691, 5.20310203439366722656185405504, 5.46867620700542721019467214194
