| L(s) = 1 | + 3.11·3-s + 5-s + 4.50·7-s + 6.72·9-s − 4.33·11-s + 3.72·13-s + 3.11·15-s + 1.11·17-s − 4.50·19-s + 14.0·21-s − 23-s + 25-s + 11.6·27-s + 8.23·29-s + 1.72·31-s − 13.5·33-s + 4.50·35-s + 0.781·37-s + 11.6·39-s + 3.90·41-s − 8·43-s + 6.72·45-s − 11.4·47-s + 13.3·49-s + 3.49·51-s + 6·53-s − 4.33·55-s + ⋯ |
| L(s) = 1 | + 1.80·3-s + 0.447·5-s + 1.70·7-s + 2.24·9-s − 1.30·11-s + 1.03·13-s + 0.805·15-s + 0.271·17-s − 1.03·19-s + 3.06·21-s − 0.208·23-s + 0.200·25-s + 2.23·27-s + 1.52·29-s + 0.310·31-s − 2.35·33-s + 0.762·35-s + 0.128·37-s + 1.86·39-s + 0.609·41-s − 1.21·43-s + 1.00·45-s − 1.67·47-s + 1.90·49-s + 0.488·51-s + 0.824·53-s − 0.584·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.755031156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.755031156\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 3.11T + 3T^{2} \) |
| 7 | \( 1 - 4.50T + 7T^{2} \) |
| 11 | \( 1 + 4.33T + 11T^{2} \) |
| 13 | \( 1 - 3.72T + 13T^{2} \) |
| 17 | \( 1 - 1.11T + 17T^{2} \) |
| 19 | \( 1 + 4.50T + 19T^{2} \) |
| 29 | \( 1 - 8.23T + 29T^{2} \) |
| 31 | \( 1 - 1.72T + 31T^{2} \) |
| 37 | \( 1 - 0.781T + 37T^{2} \) |
| 41 | \( 1 - 3.90T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 2.23T + 59T^{2} \) |
| 61 | \( 1 + 3.55T + 61T^{2} \) |
| 67 | \( 1 + 2.43T + 67T^{2} \) |
| 71 | \( 1 - 7.11T + 71T^{2} \) |
| 73 | \( 1 + 9.45T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 + 2.78T + 83T^{2} \) |
| 89 | \( 1 + 7.69T + 89T^{2} \) |
| 97 | \( 1 + 0.642T + 97T^{2} \) |
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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
−8.089790338239970761055291195944, −7.59125011680840384729518130075, −6.70383054537346097592806818639, −5.74943822753079013863137390721, −4.78355638153218370824749000126, −4.39353069738290849511257649550, −3.36547919831496856033373737317, −2.60441006045526280641676364045, −1.95173966026119337589501102907, −1.25211897208122148185778082900,
1.25211897208122148185778082900, 1.95173966026119337589501102907, 2.60441006045526280641676364045, 3.36547919831496856033373737317, 4.39353069738290849511257649550, 4.78355638153218370824749000126, 5.74943822753079013863137390721, 6.70383054537346097592806818639, 7.59125011680840384729518130075, 8.089790338239970761055291195944
