| L(s) = 1 | + 3.86·5-s − 2.44·7-s − 1.46·11-s − 4.24·13-s + 3.46·17-s − 7.46·19-s + 2.82·23-s + 9.92·25-s + 8.76·29-s − 7.34·31-s − 9.46·35-s − 0.656·37-s − 4.53·41-s + 3.46·43-s − 2.82·47-s − 1.00·49-s − 4.62·53-s − 5.65·55-s + 13.8·59-s + 0.656·61-s − 16.3·65-s − 14.9·67-s + 6.41·71-s + 4·73-s + 3.58·77-s + 2.44·79-s + 5.46·83-s + ⋯ |
| L(s) = 1 | + 1.72·5-s − 0.925·7-s − 0.441·11-s − 1.17·13-s + 0.840·17-s − 1.71·19-s + 0.589·23-s + 1.98·25-s + 1.62·29-s − 1.31·31-s − 1.59·35-s − 0.107·37-s − 0.708·41-s + 0.528·43-s − 0.412·47-s − 0.142·49-s − 0.634·53-s − 0.762·55-s + 1.80·59-s + 0.0840·61-s − 2.03·65-s − 1.82·67-s + 0.761·71-s + 0.468·73-s + 0.408·77-s + 0.275·79-s + 0.599·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 - 3.86T + 5T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 + 1.46T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 7.46T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 8.76T + 29T^{2} \) |
| 31 | \( 1 + 7.34T + 31T^{2} \) |
| 37 | \( 1 + 0.656T + 37T^{2} \) |
| 41 | \( 1 + 4.53T + 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 4.62T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 0.656T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 - 6.41T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 2.44T + 79T^{2} \) |
| 83 | \( 1 - 5.46T + 83T^{2} \) |
| 89 | \( 1 - 4.92T + 89T^{2} \) |
| 97 | \( 1 - 1.07T + 97T^{2} \) |
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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
−7.17260408738962827757774588503, −6.52933821234577505733678569610, −6.15570095339283823200213382963, −5.24896804515026262048818415478, −4.92326270139031172080576139404, −3.74197667478715081529198557643, −2.72157033237434518888690465060, −2.37891667223088474243754825198, −1.36877668485232437639124147929, 0,
1.36877668485232437639124147929, 2.37891667223088474243754825198, 2.72157033237434518888690465060, 3.74197667478715081529198557643, 4.92326270139031172080576139404, 5.24896804515026262048818415478, 6.15570095339283823200213382963, 6.52933821234577505733678569610, 7.17260408738962827757774588503
