L(s) = 1 | + 0.188·2-s − 3-s − 1.96·4-s + 3.42·5-s − 0.188·6-s + 1.78·7-s − 0.747·8-s + 9-s + 0.645·10-s + 3.14·11-s + 1.96·12-s − 5.35·13-s + 0.336·14-s − 3.42·15-s + 3.78·16-s − 17-s + 0.188·18-s − 0.832·19-s − 6.72·20-s − 1.78·21-s + 0.593·22-s − 4.73·23-s + 0.747·24-s + 6.72·25-s − 1.01·26-s − 27-s − 3.50·28-s + ⋯ |
L(s) = 1 | + 0.133·2-s − 0.577·3-s − 0.982·4-s + 1.53·5-s − 0.0769·6-s + 0.674·7-s − 0.264·8-s + 0.333·9-s + 0.204·10-s + 0.949·11-s + 0.567·12-s − 1.48·13-s + 0.0898·14-s − 0.884·15-s + 0.947·16-s − 0.242·17-s + 0.0444·18-s − 0.190·19-s − 1.50·20-s − 0.389·21-s + 0.126·22-s − 0.986·23-s + 0.152·24-s + 1.34·25-s − 0.198·26-s − 0.192·27-s − 0.662·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 157 | \( 1 + T \) |
good | 2 | \( 1 - 0.188T + 2T^{2} \) |
| 5 | \( 1 - 3.42T + 5T^{2} \) |
| 7 | \( 1 - 1.78T + 7T^{2} \) |
| 11 | \( 1 - 3.14T + 11T^{2} \) |
| 13 | \( 1 + 5.35T + 13T^{2} \) |
| 19 | \( 1 + 0.832T + 19T^{2} \) |
| 23 | \( 1 + 4.73T + 23T^{2} \) |
| 29 | \( 1 - 1.69T + 29T^{2} \) |
| 31 | \( 1 - 5.10T + 31T^{2} \) |
| 37 | \( 1 + 3.43T + 37T^{2} \) |
| 41 | \( 1 + 6.02T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + 4.41T + 47T^{2} \) |
| 53 | \( 1 - 8.72T + 53T^{2} \) |
| 59 | \( 1 - 4.56T + 59T^{2} \) |
| 61 | \( 1 + 1.80T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 + 3.53T + 71T^{2} \) |
| 73 | \( 1 + 9.39T + 73T^{2} \) |
| 79 | \( 1 + 6.65T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 - 6.57T + 97T^{2} \) |
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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
−7.36802203062136121997665463609, −6.60913806566043783578093913467, −5.99516717417843498520475295634, −5.32018505271460631699131751424, −4.77651962748883851525719900238, −4.26283582036451084430019169691, −3.08223430931047752956731901942, −1.99502219373457835832504536010, −1.36552451734416153228407689814, 0,
1.36552451734416153228407689814, 1.99502219373457835832504536010, 3.08223430931047752956731901942, 4.26283582036451084430019169691, 4.77651962748883851525719900238, 5.32018505271460631699131751424, 5.99516717417843498520475295634, 6.60913806566043783578093913467, 7.36802203062136121997665463609