| L(s) = 1 | + 1.73·2-s − 3-s + 0.994·4-s + 2.73·5-s − 1.73·6-s − 2.97·7-s − 1.74·8-s + 9-s + 4.73·10-s + 3.39·11-s − 0.994·12-s − 0.906·13-s − 5.14·14-s − 2.73·15-s − 4.99·16-s + 17-s + 1.73·18-s − 5.19·19-s + 2.71·20-s + 2.97·21-s + 5.87·22-s − 5.06·23-s + 1.74·24-s + 2.47·25-s − 1.56·26-s − 27-s − 2.95·28-s + ⋯ |
| L(s) = 1 | + 1.22·2-s − 0.577·3-s + 0.497·4-s + 1.22·5-s − 0.706·6-s − 1.12·7-s − 0.615·8-s + 0.333·9-s + 1.49·10-s + 1.02·11-s − 0.287·12-s − 0.251·13-s − 1.37·14-s − 0.705·15-s − 1.24·16-s + 0.242·17-s + 0.407·18-s − 1.19·19-s + 0.607·20-s + 0.649·21-s + 1.25·22-s − 1.05·23-s + 0.355·24-s + 0.494·25-s − 0.307·26-s − 0.192·27-s − 0.559·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 5 | \( 1 - 2.73T + 5T^{2} \) |
| 7 | \( 1 + 2.97T + 7T^{2} \) |
| 11 | \( 1 - 3.39T + 11T^{2} \) |
| 13 | \( 1 + 0.906T + 13T^{2} \) |
| 19 | \( 1 + 5.19T + 19T^{2} \) |
| 23 | \( 1 + 5.06T + 23T^{2} \) |
| 29 | \( 1 + 0.264T + 29T^{2} \) |
| 31 | \( 1 + 5.85T + 31T^{2} \) |
| 37 | \( 1 - 8.16T + 37T^{2} \) |
| 41 | \( 1 - 6.74T + 41T^{2} \) |
| 43 | \( 1 + 7.82T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 1.47T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 9.90T + 61T^{2} \) |
| 67 | \( 1 - 8.38T + 67T^{2} \) |
| 71 | \( 1 - 9.48T + 71T^{2} \) |
| 73 | \( 1 - 1.04T + 73T^{2} \) |
| 83 | \( 1 + 8.37T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 5.96T + 97T^{2} \) |
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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
−7.971637984610795317122686698617, −6.63505776418894267961240804516, −6.41827824099522461828518497481, −5.91038770864803847938657263608, −5.17947384224040877063596394407, −4.26912602165481604481603488473, −3.64183298792404226526340089857, −2.64070007685477649114596533152, −1.69468646253274585562870632576, 0,
1.69468646253274585562870632576, 2.64070007685477649114596533152, 3.64183298792404226526340089857, 4.26912602165481604481603488473, 5.17947384224040877063596394407, 5.91038770864803847938657263608, 6.41827824099522461828518497481, 6.63505776418894267961240804516, 7.971637984610795317122686698617
