| L(s) = 1 | − 0.691·2-s − 3·3-s − 7.52·4-s − 5.77·5-s + 2.07·6-s + 9.57·7-s + 10.7·8-s + 9·9-s + 3.99·10-s − 11·11-s + 22.5·12-s + 4.33·13-s − 6.62·14-s + 17.3·15-s + 52.7·16-s − 93.4·17-s − 6.22·18-s − 19.5·19-s + 43.4·20-s − 28.7·21-s + 7.60·22-s + 105.·23-s − 32.1·24-s − 91.6·25-s − 2.99·26-s − 27·27-s − 72.0·28-s + ⋯ |
| L(s) = 1 | − 0.244·2-s − 0.577·3-s − 0.940·4-s − 0.516·5-s + 0.141·6-s + 0.517·7-s + 0.474·8-s + 0.333·9-s + 0.126·10-s − 0.301·11-s + 0.542·12-s + 0.0924·13-s − 0.126·14-s + 0.298·15-s + 0.824·16-s − 1.33·17-s − 0.0814·18-s − 0.235·19-s + 0.485·20-s − 0.298·21-s + 0.0737·22-s + 0.960·23-s − 0.273·24-s − 0.733·25-s − 0.0226·26-s − 0.192·27-s − 0.486·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 0.691T + 8T^{2} \) |
| 5 | \( 1 + 5.77T + 125T^{2} \) |
| 7 | \( 1 - 9.57T + 343T^{2} \) |
| 13 | \( 1 - 4.33T + 2.19e3T^{2} \) |
| 17 | \( 1 + 93.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 19.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 105.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 157.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 208.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 270.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 401.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 125.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 22.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 722.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 208.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 441.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 950.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.06e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 448.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.38e3T + 9.12e5T^{2} \) |
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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
−8.458998236518450853190081528162, −7.71681235067371123923464059584, −6.96003135633057052356206666003, −5.90534801786471506557774995393, −5.04111814763604383356386493129, −4.42368385257447062616680286186, −3.67394556152249371461909783943, −2.20424723275579649860386492708, −0.920591100130795540303372878781, 0,
0.920591100130795540303372878781, 2.20424723275579649860386492708, 3.67394556152249371461909783943, 4.42368385257447062616680286186, 5.04111814763604383356386493129, 5.90534801786471506557774995393, 6.96003135633057052356206666003, 7.71681235067371123923464059584, 8.458998236518450853190081528162
