| L(s) = 1 | + 2-s − 2·3-s + 4-s − 1.73·5-s − 2·6-s + 3.46·7-s + 8-s + 9-s − 1.73·10-s − 3.46·11-s − 2·12-s − 5·13-s + 3.46·14-s + 3.46·15-s + 16-s + 6.92·17-s + 18-s − 3.46·19-s − 1.73·20-s − 6.92·21-s − 3.46·22-s − 2·24-s − 2.00·25-s − 5·26-s + 4·27-s + 3.46·28-s − 3·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 0.5·4-s − 0.774·5-s − 0.816·6-s + 1.30·7-s + 0.353·8-s + 0.333·9-s − 0.547·10-s − 1.04·11-s − 0.577·12-s − 1.38·13-s + 0.925·14-s + 0.894·15-s + 0.250·16-s + 1.68·17-s + 0.235·18-s − 0.794·19-s − 0.387·20-s − 1.51·21-s − 0.738·22-s − 0.408·24-s − 0.400·25-s − 0.980·26-s + 0.769·27-s + 0.654·28-s − 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1058 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1058 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2T + 3T^{2} \) |
| 5 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 + 6.92T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 1.73T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 5.19T + 61T^{2} \) |
| 67 | \( 1 - 6.92T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 1.73T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−9.850686781438369311194993382652, −8.237102099255306926722837921226, −7.74043750084963593332630479891, −6.97015064478180231885711381536, −5.63772674463909215849280276568, −5.18857017639026022543413895371, −4.55338128626278063150662009436, −3.29314479584828507229991547576, −1.86254153401304686044954392695, 0,
1.86254153401304686044954392695, 3.29314479584828507229991547576, 4.55338128626278063150662009436, 5.18857017639026022543413895371, 5.63772674463909215849280276568, 6.97015064478180231885711381536, 7.74043750084963593332630479891, 8.237102099255306926722837921226, 9.850686781438369311194993382652
