| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s + 12-s − 2·13-s − 16-s − 6·17-s + 18-s − 4·19-s + 4·23-s + 3·24-s − 2·26-s − 27-s + 6·29-s + 8·31-s + 5·32-s − 6·34-s − 36-s − 6·37-s − 4·38-s + 2·39-s − 2·41-s − 43-s + 4·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s + 0.288·12-s − 0.554·13-s − 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.834·23-s + 0.612·24-s − 0.392·26-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.883·32-s − 1.02·34-s − 1/6·36-s − 0.986·37-s − 0.648·38-s + 0.320·39-s − 0.312·41-s − 0.152·43-s + 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.303469454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.303469454\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.50586380234968, −12.04018268346054, −11.75009592777558, −11.15038874014208, −10.71788125730200, −10.23702498911320, −9.861958989572009, −9.224639593120555, −8.796326243330456, −8.521741774708553, −7.902556094670350, −7.303531557208839, −6.637120844093665, −6.387096755566831, −6.096457107112382, −5.262268832949836, −4.832559630981824, −4.568571931718897, −4.289400200747988, −3.293330493930474, −3.169999226682440, −2.302886712463444, −1.864105087116207, −0.8895452710929455, −0.3289830398990081,
0.3289830398990081, 0.8895452710929455, 1.864105087116207, 2.302886712463444, 3.169999226682440, 3.293330493930474, 4.289400200747988, 4.568571931718897, 4.832559630981824, 5.262268832949836, 6.096457107112382, 6.387096755566831, 6.637120844093665, 7.303531557208839, 7.902556094670350, 8.521741774708553, 8.796326243330456, 9.224639593120555, 9.861958989572009, 10.23702498911320, 10.71788125730200, 11.15038874014208, 11.75009592777558, 12.04018268346054, 12.50586380234968
