L(s) = 1 | + (1.50 − 1.09i)2-s + (0.759 − 2.33i)4-s + (0.303 + 0.220i)5-s + (−0.837 − 2.57i)8-s + (−0.809 + 0.587i)9-s + 0.696·10-s + (−0.929 − 0.368i)11-s + (−0.101 + 0.0738i)13-s + (−2.09 − 1.51i)16-s + (−0.574 + 1.76i)18-s + (0.541 + 1.66i)19-s + (0.745 − 0.541i)20-s + (−1.80 + 0.462i)22-s + 1.75·23-s + (−0.265 − 0.817i)25-s + (−0.0721 + 0.222i)26-s + ⋯ |
L(s) = 1 | + (1.50 − 1.09i)2-s + (0.759 − 2.33i)4-s + (0.303 + 0.220i)5-s + (−0.837 − 2.57i)8-s + (−0.809 + 0.587i)9-s + 0.696·10-s + (−0.929 − 0.368i)11-s + (−0.101 + 0.0738i)13-s + (−2.09 − 1.51i)16-s + (−0.574 + 1.76i)18-s + (0.541 + 1.66i)19-s + (0.745 − 0.541i)20-s + (−1.80 + 0.462i)22-s + 1.75·23-s + (−0.265 − 0.817i)25-s + (−0.0721 + 0.222i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0864 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0864 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.070771033\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.070771033\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (0.929 + 0.368i)T \) |
| 79 | \( 1 + (0.809 - 0.587i)T \) |
good | 2 | \( 1 + (-1.50 + 1.09i)T + (0.309 - 0.951i)T^{2} \) |
| 3 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 5 | \( 1 + (-0.303 - 0.220i)T + (0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (0.101 - 0.0738i)T + (0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.541 - 1.66i)T + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 - 1.75T + T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (1.56 - 1.13i)T + (0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + 1.98T + T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.450 + 1.38i)T + (-0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + 1.27T + T^{2} \) |
| 97 | \( 1 + (-0.688 + 0.500i)T + (0.309 - 0.951i)T^{2} \) |
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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
−10.61332718369721933352943670062, −9.759731170792491884811146737136, −8.573037862198494881396160451612, −7.39710607492159258581359783290, −6.15089465284597564583483868252, −5.45660386630276830983462467576, −4.86302319880267332169176033174, −3.49127957256481496498483528100, −2.83823165263871363299504520365, −1.74414356698575704406689491610,
2.62749725370649178838124449622, 3.38787984440629311350627760376, 4.72127230804725142268392633601, 5.29786896521867999543026712822, 6.02353261204997811092190208082, 7.09825669319460161664264086280, 7.55674184107818684607840633335, 8.757895607395666125557422206707, 9.411891648321095109960939390888, 11.02428468453140265514496746910