| L(s) = 1 | + 2-s + 2.20·3-s + 4-s − 5-s + 2.20·6-s − 1.27·7-s + 8-s + 1.85·9-s − 10-s − 4.76·11-s + 2.20·12-s − 13-s − 1.27·14-s − 2.20·15-s + 16-s − 4.85·17-s + 1.85·18-s − 0.279·19-s − 20-s − 2.80·21-s − 4.76·22-s − 7.92·23-s + 2.20·24-s + 25-s − 26-s − 2.52·27-s − 1.27·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.27·3-s + 0.5·4-s − 0.447·5-s + 0.899·6-s − 0.481·7-s + 0.353·8-s + 0.617·9-s − 0.316·10-s − 1.43·11-s + 0.635·12-s − 0.277·13-s − 0.340·14-s − 0.568·15-s + 0.250·16-s − 1.17·17-s + 0.436·18-s − 0.0640·19-s − 0.223·20-s − 0.612·21-s − 1.01·22-s − 1.65·23-s + 0.449·24-s + 0.200·25-s − 0.196·26-s − 0.486·27-s − 0.240·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 - 2.20T + 3T^{2} \) |
| 7 | \( 1 + 1.27T + 7T^{2} \) |
| 11 | \( 1 + 4.76T + 11T^{2} \) |
| 17 | \( 1 + 4.85T + 17T^{2} \) |
| 19 | \( 1 + 0.279T + 19T^{2} \) |
| 23 | \( 1 + 7.92T + 23T^{2} \) |
| 29 | \( 1 - 8.43T + 29T^{2} \) |
| 37 | \( 1 + 0.178T + 37T^{2} \) |
| 41 | \( 1 + 6.84T + 41T^{2} \) |
| 43 | \( 1 - 1.98T + 43T^{2} \) |
| 47 | \( 1 + 2.28T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 + 0.269T + 61T^{2} \) |
| 67 | \( 1 + 4.56T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 0.204T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 + 2.69T + 97T^{2} \) |
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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
−8.155614300392949686990048283621, −7.44978309975670585238756592186, −6.65951180589418888232556395808, −5.84680813133513507289239141299, −4.82555450597235211404389325370, −4.19181895865006803017737526830, −3.27395502478517877399459117605, −2.68170552120062259073956686256, −1.99940205976104445564109963545, 0,
1.99940205976104445564109963545, 2.68170552120062259073956686256, 3.27395502478517877399459117605, 4.19181895865006803017737526830, 4.82555450597235211404389325370, 5.84680813133513507289239141299, 6.65951180589418888232556395808, 7.44978309975670585238756592186, 8.155614300392949686990048283621
