| L(s) = 1 | + 3-s + 4·7-s + 9-s − 6·13-s − 2·17-s + 6·19-s + 4·21-s + 6·23-s + 27-s + 8·29-s + 8·31-s − 10·37-s − 6·39-s − 6·41-s + 4·43-s + 2·47-s + 9·49-s − 2·51-s + 6·53-s + 6·57-s − 12·59-s − 14·61-s + 4·63-s + 4·67-s + 6·69-s + 8·71-s − 4·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s − 1.66·13-s − 0.485·17-s + 1.37·19-s + 0.872·21-s + 1.25·23-s + 0.192·27-s + 1.48·29-s + 1.43·31-s − 1.64·37-s − 0.960·39-s − 0.937·41-s + 0.609·43-s + 0.291·47-s + 9/7·49-s − 0.280·51-s + 0.824·53-s + 0.794·57-s − 1.56·59-s − 1.79·61-s + 0.503·63-s + 0.488·67-s + 0.722·69-s + 0.949·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.311640414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.311640414\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−7.71757218347276386812128877766, −7.18744745950237850510309399337, −6.53969734023894710858779170584, −5.27344026076578266733817952618, −4.89976899892102779162800521198, −4.45546316688293925777219238802, −3.23452065310677781858665703876, −2.62072433313487456860712288425, −1.78586166319718233103174536497, −0.878903408044518608452260109367,
0.878903408044518608452260109367, 1.78586166319718233103174536497, 2.62072433313487456860712288425, 3.23452065310677781858665703876, 4.45546316688293925777219238802, 4.89976899892102779162800521198, 5.27344026076578266733817952618, 6.53969734023894710858779170584, 7.18744745950237850510309399337, 7.71757218347276386812128877766
