| L(s) = 1 | − 3-s − 2·4-s − 2·7-s + 9-s + 3·11-s + 2·12-s − 2·13-s + 4·16-s + 2·19-s + 2·21-s − 3·23-s − 27-s + 4·28-s − 29-s + 8·31-s − 3·33-s − 2·36-s + 37-s + 2·39-s − 3·41-s + 43-s − 6·44-s + 6·47-s − 4·48-s − 3·49-s + 4·52-s + 3·53-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.755·7-s + 1/3·9-s + 0.904·11-s + 0.577·12-s − 0.554·13-s + 16-s + 0.458·19-s + 0.436·21-s − 0.625·23-s − 0.192·27-s + 0.755·28-s − 0.185·29-s + 1.43·31-s − 0.522·33-s − 1/3·36-s + 0.164·37-s + 0.320·39-s − 0.468·41-s + 0.152·43-s − 0.904·44-s + 0.875·47-s − 0.577·48-s − 3/7·49-s + 0.554·52-s + 0.412·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−8.817096332895810205637144675074, −7.951207448871910567724918435900, −7.03366341241257287386243505120, −6.23395885448189709604270880699, −5.52849838133555747553123337967, −4.57402872833227005419321034189, −3.93897515331318142053904300882, −2.90240275077052864182688584606, −1.26264404654329734983920846065, 0,
1.26264404654329734983920846065, 2.90240275077052864182688584606, 3.93897515331318142053904300882, 4.57402872833227005419321034189, 5.52849838133555747553123337967, 6.23395885448189709604270880699, 7.03366341241257287386243505120, 7.951207448871910567724918435900, 8.817096332895810205637144675074
