| L(s) = 1 | + 0.732·3-s − 5-s − 2.46·9-s − 2·11-s + 2.73·13-s − 0.732·15-s + 0.535·17-s − 19-s − 5.46·23-s + 25-s − 4·27-s + 3.46·29-s − 4·31-s − 1.46·33-s − 9.66·37-s + 2·39-s + 7.46·41-s − 10.9·43-s + 2.46·45-s − 10.9·47-s − 7·49-s + 0.392·51-s + 5.66·53-s + 2·55-s − 0.732·57-s + 5.46·59-s − 13.4·61-s + ⋯ |
| L(s) = 1 | + 0.422·3-s − 0.447·5-s − 0.821·9-s − 0.603·11-s + 0.757·13-s − 0.189·15-s + 0.129·17-s − 0.229·19-s − 1.13·23-s + 0.200·25-s − 0.769·27-s + 0.643·29-s − 0.718·31-s − 0.254·33-s − 1.58·37-s + 0.320·39-s + 1.16·41-s − 1.66·43-s + 0.367·45-s − 1.59·47-s − 49-s + 0.0549·51-s + 0.777·53-s + 0.269·55-s − 0.0969·57-s + 0.711·59-s − 1.72·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 2.73T + 13T^{2} \) |
| 17 | \( 1 - 0.535T + 17T^{2} \) |
| 23 | \( 1 + 5.46T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 9.66T + 37T^{2} \) |
| 41 | \( 1 - 7.46T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 5.66T + 53T^{2} \) |
| 59 | \( 1 - 5.46T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 6.19T + 67T^{2} \) |
| 71 | \( 1 - 2.92T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 1.46T + 83T^{2} \) |
| 89 | \( 1 - 3.46T + 89T^{2} \) |
| 97 | \( 1 - 1.66T + 97T^{2} \) |
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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
−8.839582248298365309855643982227, −8.305123563021466750955681542974, −7.70526956241954638445114867968, −6.62366761515531442503988034032, −5.79003783707328946048839339413, −4.88300789911198665801164005072, −3.73870051374215689690157969200, −3.03868801356420318442004832131, −1.82964360229075509003421852102, 0,
1.82964360229075509003421852102, 3.03868801356420318442004832131, 3.73870051374215689690157969200, 4.88300789911198665801164005072, 5.79003783707328946048839339413, 6.62366761515531442503988034032, 7.70526956241954638445114867968, 8.305123563021466750955681542974, 8.839582248298365309855643982227
