| L(s) = 1 | + 2-s − 2.73·3-s + 4-s + 1.73·5-s − 2.73·6-s − 2·7-s + 8-s + 4.46·9-s + 1.73·10-s + 4.73·11-s − 2.73·12-s − 3.46·13-s − 2·14-s − 4.73·15-s + 16-s − 7.73·17-s + 4.46·18-s + 1.26·19-s + 1.73·20-s + 5.46·21-s + 4.73·22-s − 4.73·23-s − 2.73·24-s − 2.00·25-s − 3.46·26-s − 3.99·27-s − 2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.57·3-s + 0.5·4-s + 0.774·5-s − 1.11·6-s − 0.755·7-s + 0.353·8-s + 1.48·9-s + 0.547·10-s + 1.42·11-s − 0.788·12-s − 0.960·13-s − 0.534·14-s − 1.22·15-s + 0.250·16-s − 1.87·17-s + 1.05·18-s + 0.290·19-s + 0.387·20-s + 1.19·21-s + 1.00·22-s − 0.986·23-s − 0.557·24-s − 0.400·25-s − 0.679·26-s − 0.769·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 7.73T + 17T^{2} \) |
| 19 | \( 1 - 1.26T + 19T^{2} \) |
| 23 | \( 1 + 4.73T + 23T^{2} \) |
| 29 | \( 1 - 8.66T + 29T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 41 | \( 1 + 9.92T + 41T^{2} \) |
| 43 | \( 1 - 0.928T + 43T^{2} \) |
| 47 | \( 1 - 4.73T + 47T^{2} \) |
| 53 | \( 1 - 2.53T + 53T^{2} \) |
| 59 | \( 1 + 2.53T + 59T^{2} \) |
| 61 | \( 1 + 1.73T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 + 5.66T + 83T^{2} \) |
| 89 | \( 1 + 6.80T + 89T^{2} \) |
| 97 | \( 1 + 7.73T + 97T^{2} \) |
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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
−8.510623138326621516476805013412, −7.01225481852326724150114620206, −6.65607482395423267467531681286, −6.17231498700478138211490893016, −5.43641703514560411782166993012, −4.60279281571557584904701699579, −3.97802021878727416133530663380, −2.58886391109544257098795459155, −1.51727124795067290893734406995, 0,
1.51727124795067290893734406995, 2.58886391109544257098795459155, 3.97802021878727416133530663380, 4.60279281571557584904701699579, 5.43641703514560411782166993012, 6.17231498700478138211490893016, 6.65607482395423267467531681286, 7.01225481852326724150114620206, 8.510623138326621516476805013412
