L(s) = 1 | − 0.644·2-s + 1.56·3-s − 1.58·4-s + 5-s − 1.01·6-s + 2.57·7-s + 2.31·8-s − 0.541·9-s − 0.644·10-s − 0.484·11-s − 2.48·12-s + 0.882·13-s − 1.66·14-s + 1.56·15-s + 1.67·16-s − 17-s + 0.348·18-s − 0.647·19-s − 1.58·20-s + 4.04·21-s + 0.312·22-s − 4.65·23-s + 3.62·24-s + 25-s − 0.568·26-s − 5.55·27-s − 4.08·28-s + ⋯ |
L(s) = 1 | − 0.455·2-s + 0.905·3-s − 0.792·4-s + 0.447·5-s − 0.412·6-s + 0.973·7-s + 0.816·8-s − 0.180·9-s − 0.203·10-s − 0.146·11-s − 0.717·12-s + 0.244·13-s − 0.443·14-s + 0.404·15-s + 0.419·16-s − 0.242·17-s + 0.0822·18-s − 0.148·19-s − 0.354·20-s + 0.881·21-s + 0.0665·22-s − 0.971·23-s + 0.739·24-s + 0.200·25-s − 0.111·26-s − 1.06·27-s − 0.771·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 71 | \( 1 - T \) |
good | 2 | \( 1 + 0.644T + 2T^{2} \) |
| 3 | \( 1 - 1.56T + 3T^{2} \) |
| 7 | \( 1 - 2.57T + 7T^{2} \) |
| 11 | \( 1 + 0.484T + 11T^{2} \) |
| 13 | \( 1 - 0.882T + 13T^{2} \) |
| 19 | \( 1 + 0.647T + 19T^{2} \) |
| 23 | \( 1 + 4.65T + 23T^{2} \) |
| 29 | \( 1 + 2.24T + 29T^{2} \) |
| 31 | \( 1 + 8.97T + 31T^{2} \) |
| 37 | \( 1 - 8.96T + 37T^{2} \) |
| 41 | \( 1 + 5.61T + 41T^{2} \) |
| 43 | \( 1 + 7.22T + 43T^{2} \) |
| 47 | \( 1 - 2.17T + 47T^{2} \) |
| 53 | \( 1 + 5.55T + 53T^{2} \) |
| 59 | \( 1 + 4.25T + 59T^{2} \) |
| 61 | \( 1 + 9.10T + 61T^{2} \) |
| 67 | \( 1 + 1.56T + 67T^{2} \) |
| 73 | \( 1 + 1.70T + 73T^{2} \) |
| 79 | \( 1 + 5.54T + 79T^{2} \) |
| 83 | \( 1 - 9.82T + 83T^{2} \) |
| 89 | \( 1 - 3.13T + 89T^{2} \) |
| 97 | \( 1 + 7.65T + 97T^{2} \) |
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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
−7.917114339519067701412253236675, −7.46527668475872915395646299770, −6.25662714829356936653624639820, −5.49102031642207611914879609790, −4.78761368204780109529726511365, −4.01162442815632608023705120503, −3.22950286478458943522982856880, −2.09527382142285007070371339052, −1.51599214309762675575598678235, 0,
1.51599214309762675575598678235, 2.09527382142285007070371339052, 3.22950286478458943522982856880, 4.01162442815632608023705120503, 4.78761368204780109529726511365, 5.49102031642207611914879609790, 6.25662714829356936653624639820, 7.46527668475872915395646299770, 7.917114339519067701412253236675