L(s) = 1 | + (0.0822 − 0.165i)2-s + (0.582 + 0.770i)4-s + (0.932 + 0.361i)7-s + (0.356 − 0.0666i)8-s + (0.0922 − 0.995i)9-s + (−0.537 − 0.711i)11-s + (0.136 − 0.124i)14-s + (−0.246 + 0.864i)16-s + (−0.156 − 0.0971i)18-s + (−0.161 + 0.0302i)22-s + (0.149 − 0.526i)23-s + (−0.602 + 0.798i)25-s + (0.264 + 0.929i)28-s + (−0.243 + 0.857i)29-s + (0.390 + 0.356i)32-s + ⋯ |
L(s) = 1 | + (0.0822 − 0.165i)2-s + (0.582 + 0.770i)4-s + (0.932 + 0.361i)7-s + (0.356 − 0.0666i)8-s + (0.0922 − 0.995i)9-s + (−0.537 − 0.711i)11-s + (0.136 − 0.124i)14-s + (−0.246 + 0.864i)16-s + (−0.156 − 0.0971i)18-s + (−0.161 + 0.0302i)22-s + (0.149 − 0.526i)23-s + (−0.602 + 0.798i)25-s + (0.264 + 0.929i)28-s + (−0.243 + 0.857i)29-s + (0.390 + 0.356i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 959 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 959 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.282900739\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.282900739\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.932 - 0.361i)T \) |
| 137 | \( 1 - T \) |
good | 2 | \( 1 + (-0.0822 + 0.165i)T + (-0.602 - 0.798i)T^{2} \) |
| 3 | \( 1 + (-0.0922 + 0.995i)T^{2} \) |
| 5 | \( 1 + (0.602 - 0.798i)T^{2} \) |
| 11 | \( 1 + (0.537 + 0.711i)T + (-0.273 + 0.961i)T^{2} \) |
| 13 | \( 1 + (-0.739 - 0.673i)T^{2} \) |
| 17 | \( 1 + (-0.932 - 0.361i)T^{2} \) |
| 19 | \( 1 + (-0.445 - 0.895i)T^{2} \) |
| 23 | \( 1 + (-0.149 + 0.526i)T + (-0.850 - 0.526i)T^{2} \) |
| 29 | \( 1 + (0.243 - 0.857i)T + (-0.850 - 0.526i)T^{2} \) |
| 31 | \( 1 + (-0.445 + 0.895i)T^{2} \) |
| 37 | \( 1 + 1.20T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + (-0.465 - 0.288i)T + (0.445 + 0.895i)T^{2} \) |
| 47 | \( 1 + (0.982 + 0.183i)T^{2} \) |
| 53 | \( 1 + (1.25 + 0.778i)T + (0.445 + 0.895i)T^{2} \) |
| 59 | \( 1 + (0.982 + 0.183i)T^{2} \) |
| 61 | \( 1 + (0.982 - 0.183i)T^{2} \) |
| 67 | \( 1 + (-0.172 - 0.0666i)T + (0.739 + 0.673i)T^{2} \) |
| 71 | \( 1 + (-0.726 + 0.961i)T + (-0.273 - 0.961i)T^{2} \) |
| 73 | \( 1 + (-0.739 + 0.673i)T^{2} \) |
| 79 | \( 1 + (1.25 + 1.14i)T + (0.0922 + 0.995i)T^{2} \) |
| 83 | \( 1 + (-0.932 + 0.361i)T^{2} \) |
| 89 | \( 1 + (0.602 - 0.798i)T^{2} \) |
| 97 | \( 1 + (0.273 - 0.961i)T^{2} \) |
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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
−10.50675769651923813705392462036, −9.245201668131820078890990144288, −8.525063682180666107484020450572, −7.79422288114880171478846021902, −6.94338947054700591180935217812, −5.99345116879002379148107133150, −4.97584374981650875220125180346, −3.77839171537650792828913882031, −2.95641208187315200095737973134, −1.68463116918141560616737166162,
1.63131935126081512579444194753, 2.45758340788483198912308802544, 4.26009337931226431059781600916, 5.05329065982907345844057633324, 5.74957524608088118547386484332, 6.96863361869093115400458131166, 7.62033310248967554702830124090, 8.285925119650276255958621088237, 9.639128296775415486604777364679, 10.31500421753947949316196159810