| L(s) = 1 | + 2-s − 2.61·3-s + 4-s − 3·5-s − 2.61·6-s − 1.38·7-s + 8-s + 3.85·9-s − 3·10-s − 2.61·11-s − 2.61·12-s − 1.76·13-s − 1.38·14-s + 7.85·15-s + 16-s − 5·17-s + 3.85·18-s − 4.38·19-s − 3·20-s + 3.61·21-s − 2.61·22-s − 1.76·23-s − 2.61·24-s + 4·25-s − 1.76·26-s − 2.23·27-s − 1.38·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.51·3-s + 0.5·4-s − 1.34·5-s − 1.06·6-s − 0.522·7-s + 0.353·8-s + 1.28·9-s − 0.948·10-s − 0.789·11-s − 0.755·12-s − 0.489·13-s − 0.369·14-s + 2.02·15-s + 0.250·16-s − 1.21·17-s + 0.908·18-s − 1.00·19-s − 0.670·20-s + 0.789·21-s − 0.558·22-s − 0.367·23-s − 0.534·24-s + 0.800·25-s − 0.345·26-s − 0.430·27-s − 0.261·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3023 | \( 1+O(T) \) |
| good | 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 + 3T + 5T^{2} \) |
| 7 | \( 1 + 1.38T + 7T^{2} \) |
| 11 | \( 1 + 2.61T + 11T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 19 | \( 1 + 4.38T + 19T^{2} \) |
| 23 | \( 1 + 1.76T + 23T^{2} \) |
| 29 | \( 1 + 1.23T + 29T^{2} \) |
| 31 | \( 1 + 4.47T + 31T^{2} \) |
| 37 | \( 1 - 1.47T + 37T^{2} \) |
| 41 | \( 1 + 12.0T + 41T^{2} \) |
| 43 | \( 1 + 1.85T + 43T^{2} \) |
| 47 | \( 1 + 7.47T + 47T^{2} \) |
| 53 | \( 1 - 1.61T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 + 9.09T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 9.70T + 71T^{2} \) |
| 73 | \( 1 + 8.94T + 73T^{2} \) |
| 79 | \( 1 - 9.47T + 79T^{2} \) |
| 83 | \( 1 - 4.94T + 83T^{2} \) |
| 89 | \( 1 + 4.47T + 89T^{2} \) |
| 97 | \( 1 - 2.47T + 97T^{2} \) |
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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
−7.10116666872104284257900660537, −6.54582661909121088522910605655, −5.97644727298919479394976097139, −4.98666365586675883734657485477, −4.66820644242868404056139431046, −3.87670975025941844630570564584, −3.03223572917756441375214731217, −1.84780794084992231694967041801, 0, 0,
1.84780794084992231694967041801, 3.03223572917756441375214731217, 3.87670975025941844630570564584, 4.66820644242868404056139431046, 4.98666365586675883734657485477, 5.97644727298919479394976097139, 6.54582661909121088522910605655, 7.10116666872104284257900660537
