| L(s) = 1 | − 1.27·2-s − 0.370·4-s + 4.09·5-s + 7-s + 3.02·8-s − 5.22·10-s − 4.39·13-s − 1.27·14-s − 3.12·16-s − 4.19·17-s − 1.24·19-s − 1.51·20-s − 4.97·23-s + 11.7·25-s + 5.60·26-s − 0.370·28-s + 1.93·29-s − 1.56·31-s − 2.06·32-s + 5.35·34-s + 4.09·35-s − 0.716·37-s + 1.59·38-s + 12.3·40-s − 4.80·41-s + 1.35·43-s + 6.34·46-s + ⋯ |
| L(s) = 1 | − 0.902·2-s − 0.185·4-s + 1.82·5-s + 0.377·7-s + 1.06·8-s − 1.65·10-s − 1.21·13-s − 0.341·14-s − 0.780·16-s − 1.01·17-s − 0.286·19-s − 0.338·20-s − 1.03·23-s + 2.34·25-s + 1.09·26-s − 0.0699·28-s + 0.359·29-s − 0.280·31-s − 0.365·32-s + 0.917·34-s + 0.691·35-s − 0.117·37-s + 0.258·38-s + 1.95·40-s − 0.750·41-s + 0.206·43-s + 0.935·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.27T + 2T^{2} \) |
| 5 | \( 1 - 4.09T + 5T^{2} \) |
| 13 | \( 1 + 4.39T + 13T^{2} \) |
| 17 | \( 1 + 4.19T + 17T^{2} \) |
| 19 | \( 1 + 1.24T + 19T^{2} \) |
| 23 | \( 1 + 4.97T + 23T^{2} \) |
| 29 | \( 1 - 1.93T + 29T^{2} \) |
| 31 | \( 1 + 1.56T + 31T^{2} \) |
| 37 | \( 1 + 0.716T + 37T^{2} \) |
| 41 | \( 1 + 4.80T + 41T^{2} \) |
| 43 | \( 1 - 1.35T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 3.97T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 - 7.59T + 67T^{2} \) |
| 71 | \( 1 + 0.218T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 4.56T + 79T^{2} \) |
| 83 | \( 1 + 2.45T + 83T^{2} \) |
| 89 | \( 1 + 4.20T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−7.61175036350009876855756988457, −6.90589416171391550475565716090, −6.21558970850372994411109306027, −5.41025646114984601712550156300, −4.83823846814469640937408319538, −4.12802665624847346262170888828, −2.62127007530647430944853052976, −2.07262487854928013845830593767, −1.34505756331115863243341067528, 0,
1.34505756331115863243341067528, 2.07262487854928013845830593767, 2.62127007530647430944853052976, 4.12802665624847346262170888828, 4.83823846814469640937408319538, 5.41025646114984601712550156300, 6.21558970850372994411109306027, 6.90589416171391550475565716090, 7.61175036350009876855756988457
