| L(s) = 1 | − 2-s + 1.40·3-s + 4-s + 3.63·5-s − 1.40·6-s − 3.46·7-s − 8-s − 1.03·9-s − 3.63·10-s + 4.21·11-s + 1.40·12-s − 4.66·13-s + 3.46·14-s + 5.10·15-s + 16-s + 1.56·17-s + 1.03·18-s − 19-s + 3.63·20-s − 4.85·21-s − 4.21·22-s − 1.74·23-s − 1.40·24-s + 8.21·25-s + 4.66·26-s − 5.65·27-s − 3.46·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.810·3-s + 0.5·4-s + 1.62·5-s − 0.572·6-s − 1.30·7-s − 0.353·8-s − 0.343·9-s − 1.14·10-s + 1.27·11-s + 0.405·12-s − 1.29·13-s + 0.924·14-s + 1.31·15-s + 0.250·16-s + 0.379·17-s + 0.243·18-s − 0.229·19-s + 0.812·20-s − 1.05·21-s − 0.899·22-s − 0.364·23-s − 0.286·24-s + 1.64·25-s + 0.914·26-s − 1.08·27-s − 0.653·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 1.40T + 3T^{2} \) |
| 5 | \( 1 - 3.63T + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 4.21T + 11T^{2} \) |
| 13 | \( 1 + 4.66T + 13T^{2} \) |
| 17 | \( 1 - 1.56T + 17T^{2} \) |
| 23 | \( 1 + 1.74T + 23T^{2} \) |
| 29 | \( 1 - 3.59T + 29T^{2} \) |
| 31 | \( 1 + 7.86T + 31T^{2} \) |
| 37 | \( 1 - 1.13T + 37T^{2} \) |
| 41 | \( 1 - 1.01T + 41T^{2} \) |
| 43 | \( 1 + 6.56T + 43T^{2} \) |
| 47 | \( 1 - 3.61T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 + 6.49T + 59T^{2} \) |
| 61 | \( 1 + 0.292T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 2.17T + 71T^{2} \) |
| 73 | \( 1 - 5.79T + 73T^{2} \) |
| 79 | \( 1 + 5.89T + 79T^{2} \) |
| 83 | \( 1 + 6.45T + 83T^{2} \) |
| 89 | \( 1 + 6.34T + 89T^{2} \) |
| 97 | \( 1 - 0.532T + 97T^{2} \) |
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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
−7.42777807954507513294328632634, −6.85361005394718955324734713343, −6.11042825668094772574402979508, −5.81205059096756496079861554691, −4.70253701051658185330374722195, −3.48778829022516434707518546089, −2.92794127212382638060221186988, −2.18719027035832986546122540385, −1.48576691762480650451437680317, 0,
1.48576691762480650451437680317, 2.18719027035832986546122540385, 2.92794127212382638060221186988, 3.48778829022516434707518546089, 4.70253701051658185330374722195, 5.81205059096756496079861554691, 6.11042825668094772574402979508, 6.85361005394718955324734713343, 7.42777807954507513294328632634
