| L(s) = 1 | + 0.182·2-s + 3-s − 1.96·4-s + 1.12·5-s + 0.182·6-s − 7-s − 0.722·8-s + 9-s + 0.205·10-s + 4.01·11-s − 1.96·12-s − 0.182·14-s + 1.12·15-s + 3.80·16-s + 3.02·17-s + 0.182·18-s + 5.30·19-s − 2.22·20-s − 21-s + 0.731·22-s − 2.17·23-s − 0.722·24-s − 3.72·25-s + 27-s + 1.96·28-s + 3.53·29-s + 0.205·30-s + ⋯ |
| L(s) = 1 | + 0.128·2-s + 0.577·3-s − 0.983·4-s + 0.504·5-s + 0.0743·6-s − 0.377·7-s − 0.255·8-s + 0.333·9-s + 0.0650·10-s + 1.21·11-s − 0.567·12-s − 0.0486·14-s + 0.291·15-s + 0.950·16-s + 0.733·17-s + 0.0429·18-s + 1.21·19-s − 0.496·20-s − 0.218·21-s + 0.155·22-s − 0.453·23-s − 0.147·24-s − 0.745·25-s + 0.192·27-s + 0.371·28-s + 0.656·29-s + 0.0375·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.299423378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.299423378\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.182T + 2T^{2} \) |
| 5 | \( 1 - 1.12T + 5T^{2} \) |
| 11 | \( 1 - 4.01T + 11T^{2} \) |
| 17 | \( 1 - 3.02T + 17T^{2} \) |
| 19 | \( 1 - 5.30T + 19T^{2} \) |
| 23 | \( 1 + 2.17T + 23T^{2} \) |
| 29 | \( 1 - 3.53T + 29T^{2} \) |
| 31 | \( 1 + 4.85T + 31T^{2} \) |
| 37 | \( 1 + 6.38T + 37T^{2} \) |
| 41 | \( 1 - 4.76T + 41T^{2} \) |
| 43 | \( 1 + 2.34T + 43T^{2} \) |
| 47 | \( 1 + 9.81T + 47T^{2} \) |
| 53 | \( 1 - 2.98T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 4.70T + 61T^{2} \) |
| 67 | \( 1 + 1.77T + 67T^{2} \) |
| 71 | \( 1 - 3.24T + 71T^{2} \) |
| 73 | \( 1 + 7.80T + 73T^{2} \) |
| 79 | \( 1 - 17.4T + 79T^{2} \) |
| 83 | \( 1 + 0.705T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 1.76T + 97T^{2} \) |
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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
−8.726486917320756953227487147661, −7.935478219347537675592536017060, −7.15409305959024148137030319024, −6.23049157537679329179994922192, −5.52612554969788074519194263791, −4.72227197580591278078559683958, −3.65379974463331814656200015832, −3.39097467316838688992982801418, −1.96905325788666759024025288416, −0.899446938114066303266602622585,
0.899446938114066303266602622585, 1.96905325788666759024025288416, 3.39097467316838688992982801418, 3.65379974463331814656200015832, 4.72227197580591278078559683958, 5.52612554969788074519194263791, 6.23049157537679329179994922192, 7.15409305959024148137030319024, 7.935478219347537675592536017060, 8.726486917320756953227487147661
