L(s) = 1 | + (0.396 + 1.63i)2-s + (−1.62 + 0.838i)4-s + (−0.915 − 1.05i)8-s + (−0.786 + 0.618i)9-s + (−0.308 + 1.27i)11-s + (0.302 − 0.424i)16-s + (−1.32 − 1.04i)18-s − 2.20·22-s + (0.928 − 0.371i)23-s + (0.723 − 0.690i)25-s + (−1.61 + 1.03i)29-s + (−0.483 − 0.193i)32-s + (0.760 − 1.66i)36-s + (−0.653 + 0.513i)37-s + (0.186 − 0.215i)43-s + (−0.565 − 2.33i)44-s + ⋯ |
L(s) = 1 | + (0.396 + 1.63i)2-s + (−1.62 + 0.838i)4-s + (−0.915 − 1.05i)8-s + (−0.786 + 0.618i)9-s + (−0.308 + 1.27i)11-s + (0.302 − 0.424i)16-s + (−1.32 − 1.04i)18-s − 2.20·22-s + (0.928 − 0.371i)23-s + (0.723 − 0.690i)25-s + (−1.61 + 1.03i)29-s + (−0.483 − 0.193i)32-s + (0.760 − 1.66i)36-s + (−0.653 + 0.513i)37-s + (0.186 − 0.215i)43-s + (−0.565 − 2.33i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.003434994\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.003434994\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 23 | \( 1 + (-0.928 + 0.371i)T \) |
good | 2 | \( 1 + (-0.396 - 1.63i)T + (-0.888 + 0.458i)T^{2} \) |
| 3 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 5 | \( 1 + (-0.723 + 0.690i)T^{2} \) |
| 11 | \( 1 + (0.308 - 1.27i)T + (-0.888 - 0.458i)T^{2} \) |
| 13 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 17 | \( 1 + (0.995 - 0.0950i)T^{2} \) |
| 19 | \( 1 + (0.995 + 0.0950i)T^{2} \) |
| 29 | \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \) |
| 31 | \( 1 + (-0.928 + 0.371i)T^{2} \) |
| 37 | \( 1 + (0.653 - 0.513i)T + (0.235 - 0.971i)T^{2} \) |
| 41 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 43 | \( 1 + (-0.186 + 0.215i)T + (-0.142 - 0.989i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-1.91 - 0.182i)T + (0.981 + 0.189i)T^{2} \) |
| 59 | \( 1 + (0.327 - 0.945i)T^{2} \) |
| 61 | \( 1 + (0.786 + 0.618i)T^{2} \) |
| 67 | \( 1 + (-1.21 + 1.16i)T + (0.0475 - 0.998i)T^{2} \) |
| 71 | \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \) |
| 73 | \( 1 + (-0.580 + 0.814i)T^{2} \) |
| 79 | \( 1 + (-1.91 + 0.182i)T + (0.981 - 0.189i)T^{2} \) |
| 83 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 89 | \( 1 + (-0.928 - 0.371i)T^{2} \) |
| 97 | \( 1 + (0.959 + 0.281i)T^{2} \) |
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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
−10.37329206895269207000846086525, −9.217130852228963037719092492479, −8.600175016515324968060983898509, −7.73602045535621326925500556286, −7.11439004936926887763047518638, −6.38351858977538797384392403826, −5.17096904399776610235822935265, −5.02136554136891515659525624099, −3.75573625776204435830779725150, −2.29962949148807995480109443610,
0.809132210169102901442648499406, 2.32758831059611864819986609977, 3.28683298398694329647538297766, 3.83508222309970244323561425128, 5.21438937384576984555705347894, 5.77777192117446872490172909446, 7.07162106995749016208388498816, 8.353402238120089964412553348797, 9.081962843918597035750318637863, 9.680807653908968418533014829479