| L(s) = 1 | − 7-s + 4.44·11-s + 5.89·13-s + 4.89·17-s + 3.89·19-s + 0.449·23-s + 4.44·29-s + 6·31-s − 9.89·37-s − 5.34·41-s + 6·43-s − 4.89·47-s − 6·49-s − 0.449·53-s − 4.89·59-s − 10.7·61-s − 4.79·67-s + 16.4·71-s − 5.89·73-s − 4.44·77-s + 5.89·79-s − 12.4·83-s + 12·89-s − 5.89·91-s − 6.10·97-s + 16.8·101-s + 8.79·103-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.34·11-s + 1.63·13-s + 1.18·17-s + 0.894·19-s + 0.0937·23-s + 0.826·29-s + 1.07·31-s − 1.62·37-s − 0.835·41-s + 0.914·43-s − 0.714·47-s − 0.857·49-s − 0.0617·53-s − 0.637·59-s − 1.38·61-s − 0.586·67-s + 1.95·71-s − 0.690·73-s − 0.507·77-s + 0.663·79-s − 1.36·83-s + 1.27·89-s − 0.618·91-s − 0.619·97-s + 1.68·101-s + 0.866·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.533770479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.533770479\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 4.44T + 11T^{2} \) |
| 13 | \( 1 - 5.89T + 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 - 3.89T + 19T^{2} \) |
| 23 | \( 1 - 0.449T + 23T^{2} \) |
| 29 | \( 1 - 4.44T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 9.89T + 37T^{2} \) |
| 41 | \( 1 + 5.34T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + 4.89T + 47T^{2} \) |
| 53 | \( 1 + 0.449T + 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 + 4.79T + 67T^{2} \) |
| 71 | \( 1 - 16.4T + 71T^{2} \) |
| 73 | \( 1 + 5.89T + 73T^{2} \) |
| 79 | \( 1 - 5.89T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + 6.10T + 97T^{2} \) |
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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
−8.273971811644685542996815534532, −7.45423870474167658293566964195, −6.53598917466693260191071189271, −6.22661604730880112223671043031, −5.34796265226359739732416667861, −4.43484189419509143268660087680, −3.45330048312773063768882492278, −3.21637514335433811652015867173, −1.60765516440880997256024803241, −0.960842391730397080082309374078,
0.960842391730397080082309374078, 1.60765516440880997256024803241, 3.21637514335433811652015867173, 3.45330048312773063768882492278, 4.43484189419509143268660087680, 5.34796265226359739732416667861, 6.22661604730880112223671043031, 6.53598917466693260191071189271, 7.45423870474167658293566964195, 8.273971811644685542996815534532
