| L(s) = 1 | + 3-s − 5-s + 3·7-s + 9-s − 11-s − 13-s − 15-s − 4·17-s + 2·19-s + 3·21-s + 23-s − 4·25-s + 27-s − 9·29-s + 4·31-s − 33-s − 3·35-s − 6·37-s − 39-s + 41-s − 11·43-s − 45-s + 2·49-s − 4·51-s − 10·53-s + 55-s + 2·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 0.258·15-s − 0.970·17-s + 0.458·19-s + 0.654·21-s + 0.208·23-s − 4/5·25-s + 0.192·27-s − 1.67·29-s + 0.718·31-s − 0.174·33-s − 0.507·35-s − 0.986·37-s − 0.160·39-s + 0.156·41-s − 1.67·43-s − 0.149·45-s + 2/7·49-s − 0.560·51-s − 1.37·53-s + 0.134·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.61741158640636832116703925173, −7.21657418249183932678232120632, −6.27713751287372915489130704646, −5.27283007706094223624566646709, −4.75152145203779965125547744075, −3.97468461930372458295295770084, −3.20803046051124934660995166735, −2.18006454846049401094359749575, −1.53345781380526615411017063175, 0,
1.53345781380526615411017063175, 2.18006454846049401094359749575, 3.20803046051124934660995166735, 3.97468461930372458295295770084, 4.75152145203779965125547744075, 5.27283007706094223624566646709, 6.27713751287372915489130704646, 7.21657418249183932678232120632, 7.61741158640636832116703925173
