L(s) = 1 | − 3-s − 2·5-s − 4·7-s + 9-s − 11-s + 13-s + 2·15-s − 2·19-s + 4·21-s + 2·23-s − 25-s − 27-s + 2·29-s + 2·31-s + 33-s + 8·35-s + 12·37-s − 39-s + 6·41-s − 2·45-s + 9·49-s + 4·53-s + 2·55-s + 2·57-s − 12·59-s − 6·61-s − 4·63-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s + 0.516·15-s − 0.458·19-s + 0.872·21-s + 0.417·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.359·31-s + 0.174·33-s + 1.35·35-s + 1.97·37-s − 0.160·39-s + 0.937·41-s − 0.298·45-s + 9/7·49-s + 0.549·53-s + 0.269·55-s + 0.264·57-s − 1.56·59-s − 0.768·61-s − 0.503·63-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 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−7.66464221141891129395636718989, −6.74924572252645711292011982283, −6.30311008899772750995614820416, −5.65975225977912501579545396169, −4.61843866749663992011123657409, −4.01304365697630145126564405935, −3.24393698889058154590076171165, −2.47043567617968160513240111365, −0.919184980786326876417639193170, 0,
0.919184980786326876417639193170, 2.47043567617968160513240111365, 3.24393698889058154590076171165, 4.01304365697630145126564405935, 4.61843866749663992011123657409, 5.65975225977912501579545396169, 6.30311008899772750995614820416, 6.74924572252645711292011982283, 7.66464221141891129395636718989