| L(s) = 1 | + 0.611·5-s + 1.35·7-s + 2.57·11-s + 6.71·13-s − 5.93·17-s − 19-s − 8.68·23-s − 4.62·25-s − 1.52·29-s − 5.79·31-s + 0.829·35-s + 1.47·37-s − 0.170·41-s + 0.885·43-s − 3.44·47-s − 5.15·49-s − 10.8·53-s + 1.57·55-s − 6.76·59-s + 1.43·61-s + 4.10·65-s − 8.02·67-s − 13.3·71-s − 2.45·73-s + 3.49·77-s − 11.9·79-s − 0.426·83-s + ⋯ |
| L(s) = 1 | + 0.273·5-s + 0.513·7-s + 0.776·11-s + 1.86·13-s − 1.43·17-s − 0.229·19-s − 1.81·23-s − 0.925·25-s − 0.283·29-s − 1.04·31-s + 0.140·35-s + 0.241·37-s − 0.0265·41-s + 0.135·43-s − 0.501·47-s − 0.736·49-s − 1.48·53-s + 0.212·55-s − 0.880·59-s + 0.184·61-s + 0.508·65-s − 0.979·67-s − 1.57·71-s − 0.287·73-s + 0.398·77-s − 1.33·79-s − 0.0468·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 0.611T + 5T^{2} \) |
| 7 | \( 1 - 1.35T + 7T^{2} \) |
| 11 | \( 1 - 2.57T + 11T^{2} \) |
| 13 | \( 1 - 6.71T + 13T^{2} \) |
| 17 | \( 1 + 5.93T + 17T^{2} \) |
| 23 | \( 1 + 8.68T + 23T^{2} \) |
| 29 | \( 1 + 1.52T + 29T^{2} \) |
| 31 | \( 1 + 5.79T + 31T^{2} \) |
| 37 | \( 1 - 1.47T + 37T^{2} \) |
| 41 | \( 1 + 0.170T + 41T^{2} \) |
| 43 | \( 1 - 0.885T + 43T^{2} \) |
| 47 | \( 1 + 3.44T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 6.76T + 59T^{2} \) |
| 61 | \( 1 - 1.43T + 61T^{2} \) |
| 67 | \( 1 + 8.02T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 + 2.45T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 0.426T + 83T^{2} \) |
| 89 | \( 1 - 6.72T + 89T^{2} \) |
| 97 | \( 1 - 3.66T + 97T^{2} \) |
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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
−7.59819231892105279005484789706, −6.50507158593037763796201507973, −6.20537928176059552059031467096, −5.56909781600355751600093226755, −4.37958831617229224024911321599, −4.07293056792566131245954504722, −3.19680581252779014607001021277, −1.86683378391226245011998123851, −1.58426999882070719028365196972, 0,
1.58426999882070719028365196972, 1.86683378391226245011998123851, 3.19680581252779014607001021277, 4.07293056792566131245954504722, 4.37958831617229224024911321599, 5.56909781600355751600093226755, 6.20537928176059552059031467096, 6.50507158593037763796201507973, 7.59819231892105279005484789706
