| L(s) = 1 | + 2·7-s + 4·11-s − 2·13-s + 17-s − 4·19-s − 6·23-s − 5·25-s + 8·29-s − 2·31-s − 4·37-s + 2·41-s + 4·43-s − 12·47-s − 3·49-s − 6·53-s + 4·59-s − 4·61-s − 4·67-s + 6·71-s − 6·73-s + 8·77-s − 10·79-s − 12·83-s + 10·89-s − 4·91-s − 10·97-s + 2·101-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 1.20·11-s − 0.554·13-s + 0.242·17-s − 0.917·19-s − 1.25·23-s − 25-s + 1.48·29-s − 0.359·31-s − 0.657·37-s + 0.312·41-s + 0.609·43-s − 1.75·47-s − 3/7·49-s − 0.824·53-s + 0.520·59-s − 0.512·61-s − 0.488·67-s + 0.712·71-s − 0.702·73-s + 0.911·77-s − 1.12·79-s − 1.31·83-s + 1.05·89-s − 0.419·91-s − 1.01·97-s + 0.199·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.35498419691014888080520065584, −6.57302914600636134681802516773, −6.09075974776201592813320834649, −5.23588172519264343540352509136, −4.42079716804157445594109560028, −4.01862313798334950035216213387, −3.01660983494491490889559223378, −1.99450518293690042087486864354, −1.40144398404918363798313243548, 0,
1.40144398404918363798313243548, 1.99450518293690042087486864354, 3.01660983494491490889559223378, 4.01862313798334950035216213387, 4.42079716804157445594109560028, 5.23588172519264343540352509136, 6.09075974776201592813320834649, 6.57302914600636134681802516773, 7.35498419691014888080520065584
