| L(s) = 1 | + 2.70·2-s − 0.701·4-s + 7·7-s − 23.5·8-s − 4.01·11-s + 51.6·13-s + 18.9·14-s − 57.8·16-s + 67.5·17-s − 50.9·19-s − 10.8·22-s − 0.507·23-s + 139.·26-s − 4.91·28-s + 120.·29-s − 292.·31-s + 31.6·32-s + 182.·34-s − 144.·37-s − 137.·38-s + 57.2·41-s − 283.·43-s + 2.81·44-s − 1.37·46-s + 233.·47-s + 49·49-s − 36.2·52-s + ⋯ |
| L(s) = 1 | + 0.955·2-s − 0.0876·4-s + 0.377·7-s − 1.03·8-s − 0.110·11-s + 1.10·13-s + 0.361·14-s − 0.904·16-s + 0.963·17-s − 0.614·19-s − 0.105·22-s − 0.00460·23-s + 1.05·26-s − 0.0331·28-s + 0.768·29-s − 1.69·31-s + 0.174·32-s + 0.919·34-s − 0.644·37-s − 0.587·38-s + 0.217·41-s − 1.00·43-s + 0.00965·44-s − 0.00439·46-s + 0.725·47-s + 0.142·49-s − 0.0965·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.156637121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.156637121\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 2.70T + 8T^{2} \) |
| 11 | \( 1 + 4.01T + 1.33e3T^{2} \) |
| 13 | \( 1 - 51.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 67.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 50.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 0.507T + 1.21e4T^{2} \) |
| 29 | \( 1 - 120.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 292.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 144.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 57.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 283.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 233.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 406.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 577.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 322.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 985.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 692.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 428.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 537.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 802.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.75e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.838134311232744556932940896538, −8.447960317874745088348632089854, −7.35710330026062763203954688928, −6.37232216039071550683442061272, −5.60939127890603190249419851742, −4.97148196796488684011996480558, −3.91705952944725557138387998311, −3.39294232959851752633017615218, −2.10436174858164533486591129343, −0.75172354341462870711658817546,
0.75172354341462870711658817546, 2.10436174858164533486591129343, 3.39294232959851752633017615218, 3.91705952944725557138387998311, 4.97148196796488684011996480558, 5.60939127890603190249419851742, 6.37232216039071550683442061272, 7.35710330026062763203954688928, 8.447960317874745088348632089854, 8.838134311232744556932940896538
