L(s) = 1 | − 0.340·2-s + 3-s − 1.88·4-s − 3.93·5-s − 0.340·6-s − 0.659·7-s + 1.32·8-s + 9-s + 1.34·10-s − 1.88·12-s + 13-s + 0.224·14-s − 3.93·15-s + 3.31·16-s + 1.00·17-s − 0.340·18-s − 5.09·19-s + 7.42·20-s − 0.659·21-s + 3.10·23-s + 1.32·24-s + 10.5·25-s − 0.340·26-s + 27-s + 1.24·28-s − 0.801·29-s + 1.34·30-s + ⋯ |
L(s) = 1 | − 0.240·2-s + 0.577·3-s − 0.942·4-s − 1.76·5-s − 0.138·6-s − 0.249·7-s + 0.467·8-s + 0.333·9-s + 0.423·10-s − 0.543·12-s + 0.277·13-s + 0.0599·14-s − 1.01·15-s + 0.829·16-s + 0.243·17-s − 0.0801·18-s − 1.16·19-s + 1.65·20-s − 0.143·21-s + 0.647·23-s + 0.269·24-s + 2.10·25-s − 0.0667·26-s + 0.192·27-s + 0.234·28-s − 0.148·29-s + 0.244·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + 0.340T + 2T^{2} \) |
| 5 | \( 1 + 3.93T + 5T^{2} \) |
| 7 | \( 1 + 0.659T + 7T^{2} \) |
| 17 | \( 1 - 1.00T + 17T^{2} \) |
| 19 | \( 1 + 5.09T + 19T^{2} \) |
| 23 | \( 1 - 3.10T + 23T^{2} \) |
| 29 | \( 1 + 0.801T + 29T^{2} \) |
| 31 | \( 1 + 4.98T + 31T^{2} \) |
| 37 | \( 1 - 0.669T + 37T^{2} \) |
| 41 | \( 1 + 1.76T + 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 - 5.78T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 + 0.344T + 59T^{2} \) |
| 61 | \( 1 - 4.29T + 61T^{2} \) |
| 67 | \( 1 + 2.12T + 67T^{2} \) |
| 71 | \( 1 - 9.69T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 6.37T + 79T^{2} \) |
| 83 | \( 1 + 7.69T + 83T^{2} \) |
| 89 | \( 1 + 4.75T + 89T^{2} \) |
| 97 | \( 1 + 9.06T + 97T^{2} \) |
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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
−8.068025871938609205065021030472, −7.43849977585706052504199529255, −6.81990152972509061720791812825, −5.61302853145904156995594975052, −4.68936021919971320410151866531, −3.94158808619806245884986591224, −3.69090327056610349412611046046, −2.59534588606382397095790051920, −1.05160857787518297215750129328, 0,
1.05160857787518297215750129328, 2.59534588606382397095790051920, 3.69090327056610349412611046046, 3.94158808619806245884986591224, 4.68936021919971320410151866531, 5.61302853145904156995594975052, 6.81990152972509061720791812825, 7.43849977585706052504199529255, 8.068025871938609205065021030472