| L(s) = 1 | − 2-s − 7·4-s + 5·5-s + 36·7-s + 15·8-s − 5·10-s − 11·11-s + 2·13-s − 36·14-s + 41·16-s − 66·17-s + 140·19-s − 35·20-s + 11·22-s + 68·23-s + 25·25-s − 2·26-s − 252·28-s − 150·29-s − 128·31-s − 161·32-s + 66·34-s + 180·35-s − 314·37-s − 140·38-s + 75·40-s + 118·41-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 7/8·4-s + 0.447·5-s + 1.94·7-s + 0.662·8-s − 0.158·10-s − 0.301·11-s + 0.0426·13-s − 0.687·14-s + 0.640·16-s − 0.941·17-s + 1.69·19-s − 0.391·20-s + 0.106·22-s + 0.616·23-s + 1/5·25-s − 0.0150·26-s − 1.70·28-s − 0.960·29-s − 0.741·31-s − 0.889·32-s + 0.332·34-s + 0.869·35-s − 1.39·37-s − 0.597·38-s + 0.296·40-s + 0.449·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.815945056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.815945056\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 11 | \( 1 + p T \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 7 | \( 1 - 36 T + p^{3} T^{2} \) |
| 13 | \( 1 - 2 T + p^{3} T^{2} \) |
| 17 | \( 1 + 66 T + p^{3} T^{2} \) |
| 19 | \( 1 - 140 T + p^{3} T^{2} \) |
| 23 | \( 1 - 68 T + p^{3} T^{2} \) |
| 29 | \( 1 + 150 T + p^{3} T^{2} \) |
| 31 | \( 1 + 128 T + p^{3} T^{2} \) |
| 37 | \( 1 + 314 T + p^{3} T^{2} \) |
| 41 | \( 1 - 118 T + p^{3} T^{2} \) |
| 43 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 47 | \( 1 - 324 T + p^{3} T^{2} \) |
| 53 | \( 1 + 82 T + p^{3} T^{2} \) |
| 59 | \( 1 - 740 T + p^{3} T^{2} \) |
| 61 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 67 | \( 1 + 124 T + p^{3} T^{2} \) |
| 71 | \( 1 - 988 T + p^{3} T^{2} \) |
| 73 | \( 1 - 2 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1100 T + p^{3} T^{2} \) |
| 83 | \( 1 - 868 T + p^{3} T^{2} \) |
| 89 | \( 1 - 470 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1186 T + p^{3} T^{2} \) |
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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
−10.63179686016480019101532838854, −9.431661193510207659391346466277, −8.823362493215879982074996995273, −7.926446166710681719908048822727, −7.22281832676058035161332482752, −5.38362491586800140091214602900, −5.07298292182573506141994804979, −3.88008529787718541861556807121, −2.06389176598766317870547376036, −0.956307571665632697070405844613,
0.956307571665632697070405844613, 2.06389176598766317870547376036, 3.88008529787718541861556807121, 5.07298292182573506141994804979, 5.38362491586800140091214602900, 7.22281832676058035161332482752, 7.926446166710681719908048822727, 8.823362493215879982074996995273, 9.431661193510207659391346466277, 10.63179686016480019101532838854
