L(s) = 1 | − 0.429·2-s + 2.76·3-s − 1.81·4-s + 1.71·5-s − 1.18·6-s − 3.46·7-s + 1.63·8-s + 4.65·9-s − 0.735·10-s − 4.04·11-s − 5.02·12-s + 4.06·13-s + 1.48·14-s + 4.74·15-s + 2.92·16-s − 4.60·17-s − 2·18-s − 2.09·19-s − 3.11·20-s − 9.58·21-s + 1.73·22-s − 3.55·23-s + 4.53·24-s − 2.06·25-s − 1.74·26-s + 4.58·27-s + 6.29·28-s + ⋯ |
L(s) = 1 | − 0.303·2-s + 1.59·3-s − 0.907·4-s + 0.766·5-s − 0.485·6-s − 1.30·7-s + 0.579·8-s + 1.55·9-s − 0.232·10-s − 1.21·11-s − 1.45·12-s + 1.12·13-s + 0.397·14-s + 1.22·15-s + 0.731·16-s − 1.11·17-s − 0.471·18-s − 0.479·19-s − 0.695·20-s − 2.09·21-s + 0.369·22-s − 0.740·23-s + 0.925·24-s − 0.412·25-s − 0.342·26-s + 0.883·27-s + 1.18·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.081561962\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.081561962\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 83 | \( 1 - T \) |
good | 2 | \( 1 + 0.429T + 2T^{2} \) |
| 3 | \( 1 - 2.76T + 3T^{2} \) |
| 5 | \( 1 - 1.71T + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 + 4.04T + 11T^{2} \) |
| 13 | \( 1 - 4.06T + 13T^{2} \) |
| 17 | \( 1 + 4.60T + 17T^{2} \) |
| 19 | \( 1 + 2.09T + 19T^{2} \) |
| 23 | \( 1 + 3.55T + 23T^{2} \) |
| 29 | \( 1 + 0.181T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 - 8.34T + 37T^{2} \) |
| 41 | \( 1 - 5.35T + 41T^{2} \) |
| 43 | \( 1 + 3.68T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 + 3.24T + 53T^{2} \) |
| 59 | \( 1 - 1.97T + 59T^{2} \) |
| 61 | \( 1 - 2.39T + 61T^{2} \) |
| 67 | \( 1 + 4.50T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 - 0.572T + 79T^{2} \) |
| 89 | \( 1 + 4.48T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−13.89796330619988133600325778523, −13.40213766818186345208601826991, −12.92930577012930543185175492708, −10.39885260012894691259326947510, −9.632957247343016353934898793094, −8.813936391112972016593412793869, −7.914310921490220011058263009318, −6.14892696246517096812732182564, −4.09273863272552915947411507239, −2.61303096446109283914280089532,
2.61303096446109283914280089532, 4.09273863272552915947411507239, 6.14892696246517096812732182564, 7.914310921490220011058263009318, 8.813936391112972016593412793869, 9.632957247343016353934898793094, 10.39885260012894691259326947510, 12.92930577012930543185175492708, 13.40213766818186345208601826991, 13.89796330619988133600325778523