| L(s) = 1 | + 3-s + 5-s + 2·7-s + 9-s + 2·13-s + 15-s − 2·19-s + 2·21-s + 25-s + 27-s − 8·31-s + 2·35-s − 2·37-s + 2·39-s − 2·43-s + 45-s − 3·49-s − 6·53-s − 2·57-s − 12·59-s + 2·61-s + 2·63-s + 2·65-s − 4·67-s − 2·73-s + 75-s − 10·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.554·13-s + 0.258·15-s − 0.458·19-s + 0.436·21-s + 1/5·25-s + 0.192·27-s − 1.43·31-s + 0.338·35-s − 0.328·37-s + 0.320·39-s − 0.304·43-s + 0.149·45-s − 3/7·49-s − 0.824·53-s − 0.264·57-s − 1.56·59-s + 0.256·61-s + 0.251·63-s + 0.248·65-s − 0.488·67-s − 0.234·73-s + 0.115·75-s − 1.12·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.83305502525532, −13.43085150519020, −12.94064969660928, −12.49863009140565, −11.94979294545377, −11.28103572555884, −10.92250684768677, −10.51144557772399, −9.857879211838025, −9.414662087351189, −8.791312177684774, −8.587797236343680, −7.889772165097269, −7.508597818363402, −6.923097189653605, −6.271243395613903, −5.845733993001610, −5.160751256105796, −4.669041854030654, −4.121980931195899, −3.399093395921987, −3.001861348901528, −2.006822759105976, −1.817933833640624, −1.073249210508209, 0,
1.073249210508209, 1.817933833640624, 2.006822759105976, 3.001861348901528, 3.399093395921987, 4.121980931195899, 4.669041854030654, 5.160751256105796, 5.845733993001610, 6.271243395613903, 6.923097189653605, 7.508597818363402, 7.889772165097269, 8.587797236343680, 8.791312177684774, 9.414662087351189, 9.857879211838025, 10.51144557772399, 10.92250684768677, 11.28103572555884, 11.94979294545377, 12.49863009140565, 12.94064969660928, 13.43085150519020, 13.83305502525532
