| L(s) = 1 | − 2-s + 3-s − 6-s − 5·7-s + 8-s + 5·11-s − 2·13-s + 5·14-s − 16-s − 7·17-s − 7·19-s − 5·21-s − 5·22-s − 2·23-s + 24-s + 5·25-s + 2·26-s − 27-s − 18·29-s + 5·33-s + 7·34-s − 4·37-s + 7·38-s − 2·39-s + 8·41-s + 5·42-s + 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.408·6-s − 1.88·7-s + 0.353·8-s + 1.50·11-s − 0.554·13-s + 1.33·14-s − 1/4·16-s − 1.69·17-s − 1.60·19-s − 1.09·21-s − 1.06·22-s − 0.417·23-s + 0.204·24-s + 25-s + 0.392·26-s − 0.192·27-s − 3.34·29-s + 0.870·33-s + 1.20·34-s − 0.657·37-s + 1.13·38-s − 0.320·39-s + 1.24·41-s + 0.771·42-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5875828142\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5875828142\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + T - 52 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−11.03636167687917677423150794342, −10.65989570045205121531432061604, −9.885209671239766446295193912485, −9.483528730192644470497644354109, −9.203568794378694818929019739795, −8.920030904475143983413660957657, −8.814066218781846911788144475148, −7.85979638002388090504422929786, −7.50567041258004675739322081623, −6.85859028368445848078516211790, −6.67734413253206016608384020228, −6.09817583210124852133315326507, −5.79347851888772012721116922892, −4.69201831469839573363367913387, −4.26515255943360664778161075487, −3.64135852906212428409458204719, −3.36354974527562499121881901580, −2.16862647203008153376810629194, −2.10050992828243298710176986334, −0.46035840243235644615126640601,
0.46035840243235644615126640601, 2.10050992828243298710176986334, 2.16862647203008153376810629194, 3.36354974527562499121881901580, 3.64135852906212428409458204719, 4.26515255943360664778161075487, 4.69201831469839573363367913387, 5.79347851888772012721116922892, 6.09817583210124852133315326507, 6.67734413253206016608384020228, 6.85859028368445848078516211790, 7.50567041258004675739322081623, 7.85979638002388090504422929786, 8.814066218781846911788144475148, 8.920030904475143983413660957657, 9.203568794378694818929019739795, 9.483528730192644470497644354109, 9.885209671239766446295193912485, 10.65989570045205121531432061604, 11.03636167687917677423150794342
