L(s) = 1 | + 3-s + 3.97·5-s − 3.17·7-s + 9-s + 11-s + 13-s + 3.97·15-s − 7.80·17-s − 7.17·19-s − 3.17·21-s − 3.36·23-s + 10.7·25-s + 27-s − 7.61·29-s − 3.15·31-s + 33-s − 12.6·35-s + 2.93·37-s + 39-s + 1.64·41-s + 4.15·43-s + 3.97·45-s − 0.660·47-s + 3.06·49-s − 7.80·51-s + 0.0696·53-s + 3.97·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.77·5-s − 1.19·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s + 1.02·15-s − 1.89·17-s − 1.64·19-s − 0.692·21-s − 0.700·23-s + 2.15·25-s + 0.192·27-s − 1.41·29-s − 0.566·31-s + 0.174·33-s − 2.13·35-s + 0.482·37-s + 0.160·39-s + 0.256·41-s + 0.633·43-s + 0.592·45-s − 0.0963·47-s + 0.438·49-s − 1.09·51-s + 0.00957·53-s + 0.535·55-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 | \( 1 - 3.97T + 5T^{2} \) |
| 7 | \( 1 + 3.17T + 7T^{2} \) |
| 17 | \( 1 + 7.80T + 17T^{2} \) |
| 19 | \( 1 + 7.17T + 19T^{2} \) |
| 23 | \( 1 + 3.36T + 23T^{2} \) |
| 29 | \( 1 + 7.61T + 29T^{2} \) |
| 31 | \( 1 + 3.15T + 31T^{2} \) |
| 37 | \( 1 - 2.93T + 37T^{2} \) |
| 41 | \( 1 - 1.64T + 41T^{2} \) |
| 43 | \( 1 - 4.15T + 43T^{2} \) |
| 47 | \( 1 + 0.660T + 47T^{2} \) |
| 53 | \( 1 - 0.0696T + 53T^{2} \) |
| 59 | \( 1 + 8.88T + 59T^{2} \) |
| 61 | \( 1 + 5.47T + 61T^{2} \) |
| 67 | \( 1 + 5.50T + 67T^{2} \) |
| 71 | \( 1 + 2.11T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 0.965T + 89T^{2} \) |
| 97 | \( 1 - 7.31T + 97T^{2} \) |
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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
−7.49761772556142563741979952646, −6.69284306224787905548586320325, −6.16674892603518692111932810955, −5.88783007870307444912047054212, −4.63721565248165518575451355766, −3.98113670771878157020141866996, −2.95689695481874564368345763472, −2.19902167978871740264068120387, −1.71586815524691770531477982276, 0,
1.71586815524691770531477982276, 2.19902167978871740264068120387, 2.95689695481874564368345763472, 3.98113670771878157020141866996, 4.63721565248165518575451355766, 5.88783007870307444912047054212, 6.16674892603518692111932810955, 6.69284306224787905548586320325, 7.49761772556142563741979952646