| L(s) = 1 | − 1.81·3-s + 4.31·5-s − 0.690·7-s + 0.306·9-s + 2.95·11-s − 1.93·13-s − 7.85·15-s − 2.07·17-s − 3.74·19-s + 1.25·21-s − 4.34·23-s + 13.6·25-s + 4.89·27-s − 8.10·29-s + 2.80·31-s − 5.37·33-s − 2.98·35-s − 9.72·37-s + 3.51·39-s − 4.09·41-s − 3.02·43-s + 1.32·45-s − 6.71·47-s − 6.52·49-s + 3.77·51-s − 0.0484·53-s + 12.7·55-s + ⋯ |
| L(s) = 1 | − 1.04·3-s + 1.93·5-s − 0.261·7-s + 0.102·9-s + 0.891·11-s − 0.536·13-s − 2.02·15-s − 0.503·17-s − 0.859·19-s + 0.273·21-s − 0.906·23-s + 2.73·25-s + 0.942·27-s − 1.50·29-s + 0.503·31-s − 0.936·33-s − 0.504·35-s − 1.59·37-s + 0.563·39-s − 0.639·41-s − 0.461·43-s + 0.197·45-s − 0.979·47-s − 0.931·49-s + 0.528·51-s − 0.00665·53-s + 1.72·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 3 | \( 1 + 1.81T + 3T^{2} \) |
| 5 | \( 1 - 4.31T + 5T^{2} \) |
| 7 | \( 1 + 0.690T + 7T^{2} \) |
| 11 | \( 1 - 2.95T + 11T^{2} \) |
| 13 | \( 1 + 1.93T + 13T^{2} \) |
| 17 | \( 1 + 2.07T + 17T^{2} \) |
| 19 | \( 1 + 3.74T + 19T^{2} \) |
| 23 | \( 1 + 4.34T + 23T^{2} \) |
| 29 | \( 1 + 8.10T + 29T^{2} \) |
| 31 | \( 1 - 2.80T + 31T^{2} \) |
| 37 | \( 1 + 9.72T + 37T^{2} \) |
| 41 | \( 1 + 4.09T + 41T^{2} \) |
| 43 | \( 1 + 3.02T + 43T^{2} \) |
| 47 | \( 1 + 6.71T + 47T^{2} \) |
| 53 | \( 1 + 0.0484T + 53T^{2} \) |
| 59 | \( 1 + 4.50T + 59T^{2} \) |
| 61 | \( 1 + 9.62T + 61T^{2} \) |
| 67 | \( 1 - 0.964T + 67T^{2} \) |
| 71 | \( 1 + 7.76T + 71T^{2} \) |
| 73 | \( 1 - 16.4T + 73T^{2} \) |
| 79 | \( 1 - 6.83T + 79T^{2} \) |
| 83 | \( 1 - 9.15T + 83T^{2} \) |
| 89 | \( 1 + 2.36T + 89T^{2} \) |
| 97 | \( 1 + 5.25T + 97T^{2} \) |
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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
−8.254172379416559389225082712350, −6.88967162308468089938593240968, −6.47123777640085047239671795176, −5.98526210488860113235170641606, −5.26157719637322643325295393426, −4.64679306369011806624149729093, −3.37634589582849900239450055454, −2.18122377984791653742117822875, −1.55742081105053074740450401679, 0,
1.55742081105053074740450401679, 2.18122377984791653742117822875, 3.37634589582849900239450055454, 4.64679306369011806624149729093, 5.26157719637322643325295393426, 5.98526210488860113235170641606, 6.47123777640085047239671795176, 6.88967162308468089938593240968, 8.254172379416559389225082712350
