L(s) = 1 | − 2.47·2-s − 1.46·3-s + 4.11·4-s + 2.18·5-s + 3.63·6-s − 2.37·7-s − 5.24·8-s − 0.842·9-s − 5.40·10-s − 3.86·11-s − 6.05·12-s − 4.98·13-s + 5.86·14-s − 3.21·15-s + 4.73·16-s − 1.28·17-s + 2.08·18-s + 0.772·19-s + 9.00·20-s + 3.48·21-s + 9.56·22-s + 0.220·23-s + 7.70·24-s − 0.222·25-s + 12.3·26-s + 5.64·27-s − 9.76·28-s + ⋯ |
L(s) = 1 | − 1.74·2-s − 0.848·3-s + 2.05·4-s + 0.977·5-s + 1.48·6-s − 0.896·7-s − 1.85·8-s − 0.280·9-s − 1.70·10-s − 1.16·11-s − 1.74·12-s − 1.38·13-s + 1.56·14-s − 0.828·15-s + 1.18·16-s − 0.310·17-s + 0.491·18-s + 0.177·19-s + 2.01·20-s + 0.759·21-s + 2.03·22-s + 0.0458·23-s + 1.57·24-s − 0.0445·25-s + 2.41·26-s + 1.08·27-s − 1.84·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.02837040631\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02837040631\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 6037 | \( 1+O(T) \) |
good | 2 | \( 1 + 2.47T + 2T^{2} \) |
| 3 | \( 1 + 1.46T + 3T^{2} \) |
| 5 | \( 1 - 2.18T + 5T^{2} \) |
| 7 | \( 1 + 2.37T + 7T^{2} \) |
| 11 | \( 1 + 3.86T + 11T^{2} \) |
| 13 | \( 1 + 4.98T + 13T^{2} \) |
| 17 | \( 1 + 1.28T + 17T^{2} \) |
| 19 | \( 1 - 0.772T + 19T^{2} \) |
| 23 | \( 1 - 0.220T + 23T^{2} \) |
| 29 | \( 1 + 5.30T + 29T^{2} \) |
| 31 | \( 1 + 1.21T + 31T^{2} \) |
| 37 | \( 1 - 0.986T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 1.66T + 43T^{2} \) |
| 47 | \( 1 + 6.63T + 47T^{2} \) |
| 53 | \( 1 - 8.34T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 2.57T + 61T^{2} \) |
| 67 | \( 1 + 9.98T + 67T^{2} \) |
| 71 | \( 1 + 2.33T + 71T^{2} \) |
| 73 | \( 1 + 6.44T + 73T^{2} \) |
| 79 | \( 1 + 7.52T + 79T^{2} \) |
| 83 | \( 1 - 5.52T + 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 + 4.61T + 97T^{2} \) |
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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
−8.147531897534402840961608462316, −7.36864261093027024006875785894, −6.83734767691490702139673150922, −6.10126108729145845899879832323, −5.52826751312865991276114439403, −4.80289455067398032200166432993, −3.09764388133805351184701232354, −2.46625678344303361244966864671, −1.62665711890541766646982115026, −0.11472355937441904272151677508,
0.11472355937441904272151677508, 1.62665711890541766646982115026, 2.46625678344303361244966864671, 3.09764388133805351184701232354, 4.80289455067398032200166432993, 5.52826751312865991276114439403, 6.10126108729145845899879832323, 6.83734767691490702139673150922, 7.36864261093027024006875785894, 8.147531897534402840961608462316