| L(s) = 1 | + 3-s − 2.71·5-s + 4.30·7-s + 9-s + 11-s + 13-s − 2.71·15-s − 6.11·17-s + 0.300·19-s + 4.30·21-s − 1.49·23-s + 2.39·25-s + 27-s − 0.312·29-s − 4.55·31-s + 33-s − 11.6·35-s − 7.96·37-s + 39-s + 1.03·41-s − 12.4·43-s − 2.71·45-s − 13.1·47-s + 11.4·49-s − 6.11·51-s + 8.49·53-s − 2.71·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.21·5-s + 1.62·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s − 0.701·15-s − 1.48·17-s + 0.0689·19-s + 0.938·21-s − 0.312·23-s + 0.478·25-s + 0.192·27-s − 0.0579·29-s − 0.817·31-s + 0.174·33-s − 1.97·35-s − 1.30·37-s + 0.160·39-s + 0.160·41-s − 1.90·43-s − 0.405·45-s − 1.91·47-s + 1.64·49-s − 0.855·51-s + 1.16·53-s − 0.366·55-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 | \( 1 + 2.71T + 5T^{2} \) |
| 7 | \( 1 - 4.30T + 7T^{2} \) |
| 17 | \( 1 + 6.11T + 17T^{2} \) |
| 19 | \( 1 - 0.300T + 19T^{2} \) |
| 23 | \( 1 + 1.49T + 23T^{2} \) |
| 29 | \( 1 + 0.312T + 29T^{2} \) |
| 31 | \( 1 + 4.55T + 31T^{2} \) |
| 37 | \( 1 + 7.96T + 37T^{2} \) |
| 41 | \( 1 - 1.03T + 41T^{2} \) |
| 43 | \( 1 + 12.4T + 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 - 8.49T + 53T^{2} \) |
| 59 | \( 1 + 9.22T + 59T^{2} \) |
| 61 | \( 1 + 9.86T + 61T^{2} \) |
| 67 | \( 1 - 8.05T + 67T^{2} \) |
| 71 | \( 1 - 0.696T + 71T^{2} \) |
| 73 | \( 1 - 4.57T + 73T^{2} \) |
| 79 | \( 1 - 17.0T + 79T^{2} \) |
| 83 | \( 1 + 8.76T + 83T^{2} \) |
| 89 | \( 1 + 3.24T + 89T^{2} \) |
| 97 | \( 1 - 7.63T + 97T^{2} \) |
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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
−7.78862961604895649871221890622, −7.10097500076899642827962875710, −6.42835214563149820034135900193, −5.16541382322275349671743567275, −4.68430746432600184955344960742, −3.96078721519521088802134518356, −3.37327674759947881768956129413, −2.12321517995728517777674257235, −1.49762024985916399532804230257, 0,
1.49762024985916399532804230257, 2.12321517995728517777674257235, 3.37327674759947881768956129413, 3.96078721519521088802134518356, 4.68430746432600184955344960742, 5.16541382322275349671743567275, 6.42835214563149820034135900193, 7.10097500076899642827962875710, 7.78862961604895649871221890622
