| L(s) = 1 | − 2·2-s + 2·3-s + 12·5-s − 4·6-s + 8·8-s + 27·9-s − 24·10-s − 48·11-s + 112·13-s + 24·15-s − 16·16-s + 114·17-s − 54·18-s − 2·19-s + 96·22-s + 120·23-s + 16·24-s + 125·25-s − 224·26-s + 154·27-s − 108·29-s − 48·30-s − 236·31-s − 96·33-s − 228·34-s − 146·37-s + 4·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.384·3-s + 1.07·5-s − 0.272·6-s + 0.353·8-s + 9-s − 0.758·10-s − 1.31·11-s + 2.38·13-s + 0.413·15-s − 1/4·16-s + 1.62·17-s − 0.707·18-s − 0.0241·19-s + 0.930·22-s + 1.08·23-s + 0.136·24-s + 25-s − 1.68·26-s + 1.09·27-s − 0.691·29-s − 0.292·30-s − 1.36·31-s − 0.506·33-s − 1.15·34-s − 0.648·37-s + 0.0170·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.213071086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.213071086\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T - 23 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 12 T + 19 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 48 T + 973 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 56 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 114 T + 8083 T^{2} - 114 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 6855 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 120 T + 2233 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 54 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 236 T + 25905 T^{2} + 236 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 146 T - 29337 T^{2} + 146 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 126 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 376 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 12 T - 103679 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 174 T - 118601 T^{2} + 174 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 138 T - 186335 T^{2} + 138 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 380 T - 82581 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 484 T - 66507 T^{2} - 484 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 576 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 1150 T + 933483 T^{2} - 1150 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 776 T + 109137 T^{2} + 776 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 378 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 390 T - 552869 T^{2} - 390 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1330 T + p^{3} T^{2} )^{2} \) |
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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
−13.44779623812582447742930829332, −13.37267652831105180251835574175, −12.72137853165263605218661264443, −12.43388070187998076446156057238, −11.07973383950148494762162354925, −11.00895022412397452990427924593, −10.32589401020255674270607193649, −9.934656354206426949715162666662, −9.340154020472940110806975944910, −8.827683652095119099116219878102, −8.202330154278258437081232592380, −7.78175582400896839882650385764, −6.93993864312313608032232048251, −6.35751161311983565853535862198, −5.42074390602952823928037517410, −5.11461118933648415097306758072, −3.76242010308913674379068395951, −3.13294608843190552641965629398, −1.78164535812667258158828769471, −1.08630948304263023849913396543,
1.08630948304263023849913396543, 1.78164535812667258158828769471, 3.13294608843190552641965629398, 3.76242010308913674379068395951, 5.11461118933648415097306758072, 5.42074390602952823928037517410, 6.35751161311983565853535862198, 6.93993864312313608032232048251, 7.78175582400896839882650385764, 8.202330154278258437081232592380, 8.827683652095119099116219878102, 9.340154020472940110806975944910, 9.934656354206426949715162666662, 10.32589401020255674270607193649, 11.00895022412397452990427924593, 11.07973383950148494762162354925, 12.43388070187998076446156057238, 12.72137853165263605218661264443, 13.37267652831105180251835574175, 13.44779623812582447742930829332
