| L(s) = 1 | − 2.64·7-s + 11-s − 4.64·17-s + 5.29·19-s + 8.29·23-s − 5·25-s − 6.64·29-s + 1.29·31-s + 2.29·37-s + 5.93·41-s − 4.64·43-s − 4.29·47-s − 4·53-s + 7·59-s + 10.5·61-s − 13.2·67-s − 6.58·71-s − 14.5·73-s − 2.64·77-s + 11.9·79-s + 8·83-s − 2.70·89-s − 13.5·97-s − 5.35·101-s − 14.5·103-s + 10.5·107-s − 2·109-s + ⋯ |
| L(s) = 1 | − 0.999·7-s + 0.301·11-s − 1.12·17-s + 1.21·19-s + 1.72·23-s − 25-s − 1.23·29-s + 0.231·31-s + 0.376·37-s + 0.927·41-s − 0.708·43-s − 0.625·47-s − 0.549·53-s + 0.911·59-s + 1.35·61-s − 1.62·67-s − 0.781·71-s − 1.70·73-s − 0.301·77-s + 1.34·79-s + 0.878·83-s − 0.287·89-s − 1.37·97-s − 0.532·101-s − 1.43·103-s + 1.02·107-s − 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 2.64T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 4.64T + 17T^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 23 | \( 1 - 8.29T + 23T^{2} \) |
| 29 | \( 1 + 6.64T + 29T^{2} \) |
| 31 | \( 1 - 1.29T + 31T^{2} \) |
| 37 | \( 1 - 2.29T + 37T^{2} \) |
| 41 | \( 1 - 5.93T + 41T^{2} \) |
| 43 | \( 1 + 4.64T + 43T^{2} \) |
| 47 | \( 1 + 4.29T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 - 7T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 6.58T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 + 2.70T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−7.83176239421351398994892990010, −7.11866061571100045285641904523, −6.56644134774103095424577332083, −5.78553453893309239180412491488, −5.02165417874334781031430941157, −4.07565257412990917913935909007, −3.31228705090494769298892673910, −2.54875870170649145034128017817, −1.32393309398928847411831885048, 0,
1.32393309398928847411831885048, 2.54875870170649145034128017817, 3.31228705090494769298892673910, 4.07565257412990917913935909007, 5.02165417874334781031430941157, 5.78553453893309239180412491488, 6.56644134774103095424577332083, 7.11866061571100045285641904523, 7.83176239421351398994892990010
