| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s − 2·11-s − 6·13-s + 2·14-s + 16-s + 2·17-s + 2·22-s − 4·23-s + 6·26-s − 2·28-s − 8·31-s − 32-s − 2·34-s − 2·37-s − 2·41-s + 4·43-s − 2·44-s + 4·46-s − 8·47-s − 3·49-s − 6·52-s + 6·53-s + 2·56-s − 10·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.603·11-s − 1.66·13-s + 0.534·14-s + 1/4·16-s + 0.485·17-s + 0.426·22-s − 0.834·23-s + 1.17·26-s − 0.377·28-s − 1.43·31-s − 0.176·32-s − 0.342·34-s − 0.328·37-s − 0.312·41-s + 0.609·43-s − 0.301·44-s + 0.589·46-s − 1.16·47-s − 3/7·49-s − 0.832·52-s + 0.824·53-s + 0.267·56-s − 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−10.29993155809996117013598530448, −9.853706775295411456819711020422, −8.977954234037566781689876272399, −7.79436266718632723136861068704, −7.19091277442794332068165955627, −6.04674329942688851458805581356, −4.95720185791601620138850804643, −3.36984414779728341388392819787, −2.17693746121355235992569926706, 0,
2.17693746121355235992569926706, 3.36984414779728341388392819787, 4.95720185791601620138850804643, 6.04674329942688851458805581356, 7.19091277442794332068165955627, 7.79436266718632723136861068704, 8.977954234037566781689876272399, 9.853706775295411456819711020422, 10.29993155809996117013598530448
