| L(s) = 1 | + 1.73·2-s + 0.999·4-s + 1.73·5-s − 2·7-s − 1.73·8-s + 2.99·10-s + 13-s − 3.46·14-s − 5·16-s + 5.19·17-s − 2·19-s + 1.73·20-s − 3.46·23-s − 2.00·25-s + 1.73·26-s − 1.99·28-s + 1.73·29-s + 8·31-s − 5.19·32-s + 9·34-s − 3.46·35-s − 7·37-s − 3.46·38-s − 3.00·40-s − 6.92·41-s − 2·43-s − 5.99·46-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.499·4-s + 0.774·5-s − 0.755·7-s − 0.612·8-s + 0.948·10-s + 0.277·13-s − 0.925·14-s − 1.25·16-s + 1.26·17-s − 0.458·19-s + 0.387·20-s − 0.722·23-s − 0.400·25-s + 0.339·26-s − 0.377·28-s + 0.321·29-s + 1.43·31-s − 0.918·32-s + 1.54·34-s − 0.585·35-s − 1.15·37-s − 0.561·38-s − 0.474·40-s − 1.08·41-s − 0.304·43-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 5.19T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 1.73T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 5.19T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−6.97628079058080314939730537220, −6.24562481505812713463926433936, −6.06420997555128303906823363261, −5.21904369715254240316449956535, −4.67803989923594633740667786051, −3.68939238846573103703420248090, −3.27380712296299602714362207289, −2.44941844836081933712830884599, −1.47452825009956490658778358704, 0,
1.47452825009956490658778358704, 2.44941844836081933712830884599, 3.27380712296299602714362207289, 3.68939238846573103703420248090, 4.67803989923594633740667786051, 5.21904369715254240316449956535, 6.06420997555128303906823363261, 6.24562481505812713463926433936, 6.97628079058080314939730537220
