| L(s) = 1 | + 4·2-s + 8·4-s + 5·5-s + 20·10-s + 10·11-s − 24·13-s − 64·16-s − 54·17-s − 12·19-s + 40·20-s + 40·22-s + 134·23-s + 25·25-s − 96·26-s − 118·29-s + 144·31-s − 256·32-s − 216·34-s − 378·37-s − 48·38-s − 330·41-s + 204·43-s + 80·44-s + 536·46-s − 312·47-s + 100·50-s − 192·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.447·5-s + 0.632·10-s + 0.274·11-s − 0.512·13-s − 16-s − 0.770·17-s − 0.144·19-s + 0.447·20-s + 0.387·22-s + 1.21·23-s + 1/5·25-s − 0.724·26-s − 0.755·29-s + 0.834·31-s − 1.41·32-s − 1.08·34-s − 1.67·37-s − 0.204·38-s − 1.25·41-s + 0.723·43-s + 0.274·44-s + 1.71·46-s − 0.968·47-s + 0.282·50-s − 0.512·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 11 | \( 1 - 10 T + p^{3} T^{2} \) |
| 13 | \( 1 + 24 T + p^{3} T^{2} \) |
| 17 | \( 1 + 54 T + p^{3} T^{2} \) |
| 19 | \( 1 + 12 T + p^{3} T^{2} \) |
| 23 | \( 1 - 134 T + p^{3} T^{2} \) |
| 29 | \( 1 + 118 T + p^{3} T^{2} \) |
| 31 | \( 1 - 144 T + p^{3} T^{2} \) |
| 37 | \( 1 + 378 T + p^{3} T^{2} \) |
| 41 | \( 1 + 330 T + p^{3} T^{2} \) |
| 43 | \( 1 - 204 T + p^{3} T^{2} \) |
| 47 | \( 1 + 312 T + p^{3} T^{2} \) |
| 53 | \( 1 + 142 T + p^{3} T^{2} \) |
| 59 | \( 1 + 108 T + p^{3} T^{2} \) |
| 61 | \( 1 + 168 T + p^{3} T^{2} \) |
| 67 | \( 1 - 448 T + p^{3} T^{2} \) |
| 71 | \( 1 + 146 T + p^{3} T^{2} \) |
| 73 | \( 1 + 360 T + p^{3} T^{2} \) |
| 79 | \( 1 - 236 T + p^{3} T^{2} \) |
| 83 | \( 1 + 324 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1194 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1728 T + p^{3} T^{2} \) |
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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
−8.366309529963010140852569597337, −7.03490604148986916502529750701, −6.68899297872873420537983814035, −5.72011550458693280025461971692, −5.04768210831401441055457805254, −4.39837473663518529237420634246, −3.42132235408760409148481415533, −2.63717339645863695683658834157, −1.63022112123580604747147722509, 0,
1.63022112123580604747147722509, 2.63717339645863695683658834157, 3.42132235408760409148481415533, 4.39837473663518529237420634246, 5.04768210831401441055457805254, 5.72011550458693280025461971692, 6.68899297872873420537983814035, 7.03490604148986916502529750701, 8.366309529963010140852569597337
