L(s) = 1 | + 2-s + 3-s + 4-s − 1.15·5-s + 6-s − 2.62·7-s + 8-s + 9-s − 1.15·10-s − 1.79·11-s + 12-s + 13-s − 2.62·14-s − 1.15·15-s + 16-s + 3.85·17-s + 18-s + 1.79·19-s − 1.15·20-s − 2.62·21-s − 1.79·22-s − 1.50·23-s + 24-s − 3.67·25-s + 26-s + 27-s − 2.62·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.514·5-s + 0.408·6-s − 0.990·7-s + 0.353·8-s + 0.333·9-s − 0.363·10-s − 0.540·11-s + 0.288·12-s + 0.277·13-s − 0.700·14-s − 0.297·15-s + 0.250·16-s + 0.935·17-s + 0.235·18-s + 0.412·19-s − 0.257·20-s − 0.571·21-s − 0.382·22-s − 0.313·23-s + 0.204·24-s − 0.735·25-s + 0.196·26-s + 0.192·27-s − 0.495·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 + 1.15T + 5T^{2} \) |
| 7 | \( 1 + 2.62T + 7T^{2} \) |
| 11 | \( 1 + 1.79T + 11T^{2} \) |
| 17 | \( 1 - 3.85T + 17T^{2} \) |
| 19 | \( 1 - 1.79T + 19T^{2} \) |
| 23 | \( 1 + 1.50T + 23T^{2} \) |
| 29 | \( 1 + 4.09T + 29T^{2} \) |
| 31 | \( 1 + 7.08T + 31T^{2} \) |
| 37 | \( 1 - 0.639T + 37T^{2} \) |
| 41 | \( 1 - 8.05T + 41T^{2} \) |
| 43 | \( 1 + 5.61T + 43T^{2} \) |
| 47 | \( 1 - 8.43T + 47T^{2} \) |
| 53 | \( 1 + 4.93T + 53T^{2} \) |
| 59 | \( 1 - 1.09T + 59T^{2} \) |
| 61 | \( 1 + 2.73T + 61T^{2} \) |
| 67 | \( 1 + 2.82T + 67T^{2} \) |
| 71 | \( 1 - 0.437T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 + 3.04T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 7.12T + 89T^{2} \) |
| 97 | \( 1 - 2.70T + 97T^{2} \) |
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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
−7.53029330596793084819864185718, −6.88246520119634643532507315405, −5.87239128267965940095904613882, −5.57729504559234525003408593319, −4.45767158940573073320912810151, −3.75290286048747435048104745653, −3.26921420789251321968306325921, −2.54101908987783858501958633561, −1.46346008130436752583073201287, 0,
1.46346008130436752583073201287, 2.54101908987783858501958633561, 3.26921420789251321968306325921, 3.75290286048747435048104745653, 4.45767158940573073320912810151, 5.57729504559234525003408593319, 5.87239128267965940095904613882, 6.88246520119634643532507315405, 7.53029330596793084819864185718