L(s) = 1 | + (0.781 − 0.376i)3-s + (0.900 − 0.433i)5-s + (0.974 + 0.222i)7-s + (−0.153 + 0.193i)9-s + (0.541 − 0.678i)15-s + (0.846 − 0.193i)21-s + (−0.193 + 0.846i)23-s + (0.623 − 0.781i)25-s + (−0.240 + 1.05i)27-s + (0.0990 + 0.433i)29-s + (0.974 − 0.222i)35-s + (1.12 − 0.541i)41-s + (−1.40 − 0.678i)43-s + (−0.0549 + 0.240i)45-s + (−0.541 − 0.678i)47-s + ⋯ |
L(s) = 1 | + (0.781 − 0.376i)3-s + (0.900 − 0.433i)5-s + (0.974 + 0.222i)7-s + (−0.153 + 0.193i)9-s + (0.541 − 0.678i)15-s + (0.846 − 0.193i)21-s + (−0.193 + 0.846i)23-s + (0.623 − 0.781i)25-s + (−0.240 + 1.05i)27-s + (0.0990 + 0.433i)29-s + (0.974 − 0.222i)35-s + (1.12 − 0.541i)41-s + (−1.40 − 0.678i)43-s + (−0.0549 + 0.240i)45-s + (−0.541 − 0.678i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.172619980\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.172619980\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.900 + 0.433i)T \) |
| 7 | \( 1 + (-0.974 - 0.222i)T \) |
good | 3 | \( 1 + (-0.781 + 0.376i)T + (0.623 - 0.781i)T^{2} \) |
| 11 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 13 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 17 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.193 - 0.846i)T + (-0.900 - 0.433i)T^{2} \) |
| 29 | \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 41 | \( 1 + (-1.12 + 0.541i)T + (0.623 - 0.781i)T^{2} \) |
| 43 | \( 1 + (1.40 + 0.678i)T + (0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (0.541 + 0.678i)T + (-0.222 + 0.974i)T^{2} \) |
| 53 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 59 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 61 | \( 1 + (0.400 + 1.75i)T + (-0.900 + 0.433i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (1.21 - 1.52i)T + (-0.222 - 0.974i)T^{2} \) |
| 89 | \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.524476360589082793347436511566, −8.061892841089516468779160540236, −7.30644022396604783747140871052, −6.43135026214968401913863390169, −5.37030717292636650115333501018, −5.14871800271878050645648952246, −3.98353365238799565351441544197, −2.89151744844605413215845041845, −2.04208652143771402445343549783, −1.44620562140161034808365075954,
1.38981501605825960274732569610, 2.41974808595187927805432452541, 3.05268449150527049373556350159, 4.12485126140761149069327190549, 4.79867384008359893546821978306, 5.79183853560236324816882207307, 6.40185346222236560801940665356, 7.32118995906159063115277458667, 8.113134248054802747109214884831, 8.672446141628505026316622252460