| L(s) = 1 | + 2·3-s + 7-s + 9-s + 4·11-s − 6·13-s + 2·21-s − 23-s − 5·25-s − 4·27-s − 2·29-s − 2·31-s + 8·33-s − 2·37-s − 12·39-s − 6·41-s + 4·43-s + 2·47-s + 49-s − 14·53-s + 14·59-s − 12·61-s + 63-s − 4·67-s − 2·69-s − 2·73-s − 10·75-s + 4·77-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 1.66·13-s + 0.436·21-s − 0.208·23-s − 25-s − 0.769·27-s − 0.371·29-s − 0.359·31-s + 1.39·33-s − 0.328·37-s − 1.92·39-s − 0.937·41-s + 0.609·43-s + 0.291·47-s + 1/7·49-s − 1.92·53-s + 1.82·59-s − 1.53·61-s + 0.125·63-s − 0.488·67-s − 0.240·69-s − 0.234·73-s − 1.15·75-s + 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−16.93381721572475, −16.39496681831502, −15.37831921204087, −15.15214094802366, −14.40431967147312, −14.23189042194999, −13.68523964223713, −12.88722721723556, −12.24454300298356, −11.74115158949026, −11.18078464761274, −10.24161882083024, −9.614211531756728, −9.269773294776337, −8.620708591774556, −7.924350645152293, −7.455694229933261, −6.821331753807620, −5.947876138981294, −5.165935787005167, −4.355896444895042, −3.744875620122098, −2.963901416575064, −2.168873522599320, −1.552948187628666, 0,
1.552948187628666, 2.168873522599320, 2.963901416575064, 3.744875620122098, 4.355896444895042, 5.165935787005167, 5.947876138981294, 6.821331753807620, 7.455694229933261, 7.924350645152293, 8.620708591774556, 9.269773294776337, 9.614211531756728, 10.24161882083024, 11.18078464761274, 11.74115158949026, 12.24454300298356, 12.88722721723556, 13.68523964223713, 14.23189042194999, 14.40431967147312, 15.15214094802366, 15.37831921204087, 16.39496681831502, 16.93381721572475
