| L(s) = 1 | + 5·3-s − 3·5-s + 64·7-s + 27·9-s − 8·11-s + 69·13-s − 15·15-s − 19·17-s − 152·19-s + 320·21-s + 67·23-s + 125·25-s + 280·27-s − 51·29-s + 264·31-s − 40·33-s − 192·35-s − 28·37-s + 345·39-s + 413·41-s + 129·43-s − 81·45-s − 617·47-s + 2.38e3·49-s − 95·51-s − 383·53-s + 24·55-s + ⋯ |
| L(s) = 1 | + 0.962·3-s − 0.268·5-s + 3.45·7-s + 9-s − 0.219·11-s + 1.47·13-s − 0.258·15-s − 0.271·17-s − 1.83·19-s + 3.32·21-s + 0.607·23-s + 25-s + 1.99·27-s − 0.326·29-s + 1.52·31-s − 0.211·33-s − 0.927·35-s − 0.124·37-s + 1.41·39-s + 1.57·41-s + 0.457·43-s − 0.268·45-s − 1.91·47-s + 6.95·49-s − 0.260·51-s − 0.992·53-s + 0.0588·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.088956967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.088956967\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 8 p T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T - 2 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T - 116 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 32 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 69 T + 2564 T^{2} - 69 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 19 T - 4552 T^{2} + 19 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 67 T - 7678 T^{2} - 67 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 51 T - 21788 T^{2} + 51 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 132 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 14 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 413 T + 101648 T^{2} - 413 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 3 p T - 34 p^{2} T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 617 T + 276866 T^{2} + 617 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 383 T - 2188 T^{2} + 383 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 599 T + 153422 T^{2} + 599 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 217 T - 179892 T^{2} - 217 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 225 T - 250138 T^{2} + 225 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 701 T + 133490 T^{2} - 701 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 1015 T + 641208 T^{2} + 1015 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 349 T - 371238 T^{2} - 349 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 592 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1349 T + 1114832 T^{2} - 1349 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 613 T - 536904 T^{2} - 613 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.26036432125972999919275824498, −10.97685310780356334466338787819, −10.64220905982274715148150337466, −10.58751856923362114643028727782, −9.362520110642994220218523094382, −8.980104789937316191089508375190, −8.339519459360120079793588912012, −8.288208054815968172294986680614, −8.005831502212826414832625451200, −7.46757509786182678312665211443, −6.63452109727474317058702207339, −6.27839499819432467535723610429, −5.12606899310940157099800544718, −4.90487738670928548490449057449, −4.29369593265880834361066729283, −4.05993433049977019832226332017, −2.87385258008614393787372150709, −2.23337767309651687195826141930, −1.41601266644951455383634763722, −1.15643684458560569653147631725,
1.15643684458560569653147631725, 1.41601266644951455383634763722, 2.23337767309651687195826141930, 2.87385258008614393787372150709, 4.05993433049977019832226332017, 4.29369593265880834361066729283, 4.90487738670928548490449057449, 5.12606899310940157099800544718, 6.27839499819432467535723610429, 6.63452109727474317058702207339, 7.46757509786182678312665211443, 8.005831502212826414832625451200, 8.288208054815968172294986680614, 8.339519459360120079793588912012, 8.980104789937316191089508375190, 9.362520110642994220218523094382, 10.58751856923362114643028727782, 10.64220905982274715148150337466, 10.97685310780356334466338787819, 11.26036432125972999919275824498
