| L(s) = 1 | + 3-s + 5-s + 4·7-s − 2·9-s + 4·11-s + 2·13-s + 15-s + 3·17-s + 6·19-s + 4·21-s + 4·23-s + 25-s − 5·27-s + 8·29-s − 5·31-s + 4·33-s + 4·35-s + 2·37-s + 2·39-s − 2·43-s − 2·45-s − 9·47-s + 9·49-s + 3·51-s − 7·53-s + 4·55-s + 6·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.51·7-s − 2/3·9-s + 1.20·11-s + 0.554·13-s + 0.258·15-s + 0.727·17-s + 1.37·19-s + 0.872·21-s + 0.834·23-s + 1/5·25-s − 0.962·27-s + 1.48·29-s − 0.898·31-s + 0.696·33-s + 0.676·35-s + 0.328·37-s + 0.320·39-s − 0.304·43-s − 0.298·45-s − 1.31·47-s + 9/7·49-s + 0.420·51-s − 0.961·53-s + 0.539·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.890752606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.890752606\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−8.096338266982707275374041656195, −7.58758511735301198688193851631, −6.67254093436087822328926389772, −5.87544206183409107090197755957, −5.17823754228234215590972991904, −4.52456018591411714333317658048, −3.46681457680301291585675520032, −2.88549138886271943777182764159, −1.64524369868047290900184145087, −1.18454366562791312800874425388,
1.18454366562791312800874425388, 1.64524369868047290900184145087, 2.88549138886271943777182764159, 3.46681457680301291585675520032, 4.52456018591411714333317658048, 5.17823754228234215590972991904, 5.87544206183409107090197755957, 6.67254093436087822328926389772, 7.58758511735301198688193851631, 8.096338266982707275374041656195
