L(s) = 1 | + (−0.190 − 0.587i)3-s + (−0.951 + 0.309i)5-s + i·7-s + (0.5 − 0.363i)9-s + (0.809 + 0.587i)11-s + (−0.587 − 0.809i)13-s + (0.363 + 0.5i)15-s + (−0.190 + 0.587i)17-s + (−0.5 + 1.53i)19-s + (0.587 − 0.190i)21-s + (−0.587 + 0.809i)23-s + (0.809 − 0.587i)25-s + (−0.809 − 0.587i)27-s + (0.951 − 0.309i)29-s + (0.951 + 0.309i)31-s + ⋯ |
L(s) = 1 | + (−0.190 − 0.587i)3-s + (−0.951 + 0.309i)5-s + i·7-s + (0.5 − 0.363i)9-s + (0.809 + 0.587i)11-s + (−0.587 − 0.809i)13-s + (0.363 + 0.5i)15-s + (−0.190 + 0.587i)17-s + (−0.5 + 1.53i)19-s + (0.587 − 0.190i)21-s + (−0.587 + 0.809i)23-s + (0.809 − 0.587i)25-s + (−0.809 − 0.587i)27-s + (0.951 − 0.309i)29-s + (0.951 + 0.309i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8790973703\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8790973703\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.951 - 0.309i)T \) |
good | 3 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 - iT - T^{2} \) |
| 11 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 71 | \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
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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
−9.872419593208841336675240633690, −8.663252638993537674353055199584, −8.036949970493345011302227447105, −7.34599149209054378889615897648, −6.42956243227564473999446142045, −5.88710288014835531906135789299, −4.56495705478741003235026362961, −3.80547869412704237945039475921, −2.65812602483812358060101301225, −1.41812033966594888991899230291,
0.798171941341680296420044569896, 2.57799399665655814024056689356, 4.10406886351164268100080291185, 4.22332973359119128743363745230, 5.05189164040826793547765312932, 6.52347748400643249944356308688, 7.11474946581388964399105293778, 7.84710650641998997001473933154, 8.906248548653584937728424893625, 9.384386592226789053726971748527