| L(s) = 1 | − 2.63·2-s + 2.70·3-s + 4.93·4-s + 2.79·5-s − 7.12·6-s − 5.09·7-s − 7.72·8-s + 4.32·9-s − 7.34·10-s − 0.629·11-s + 13.3·12-s − 1.20·13-s + 13.4·14-s + 7.55·15-s + 10.4·16-s + 17-s − 11.3·18-s − 4.74·19-s + 13.7·20-s − 13.7·21-s + 1.65·22-s − 1.58·23-s − 20.8·24-s + 2.78·25-s + 3.16·26-s + 3.57·27-s − 25.1·28-s + ⋯ |
| L(s) = 1 | − 1.86·2-s + 1.56·3-s + 2.46·4-s + 1.24·5-s − 2.90·6-s − 1.92·7-s − 2.73·8-s + 1.44·9-s − 2.32·10-s − 0.189·11-s + 3.85·12-s − 0.333·13-s + 3.58·14-s + 1.94·15-s + 2.61·16-s + 0.242·17-s − 2.68·18-s − 1.08·19-s + 3.07·20-s − 3.00·21-s + 0.353·22-s − 0.331·23-s − 4.26·24-s + 0.557·25-s + 0.620·26-s + 0.688·27-s − 4.75·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.344918436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.344918436\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 353 | \( 1 + T \) |
| good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 3 | \( 1 - 2.70T + 3T^{2} \) |
| 5 | \( 1 - 2.79T + 5T^{2} \) |
| 7 | \( 1 + 5.09T + 7T^{2} \) |
| 11 | \( 1 + 0.629T + 11T^{2} \) |
| 13 | \( 1 + 1.20T + 13T^{2} \) |
| 19 | \( 1 + 4.74T + 19T^{2} \) |
| 23 | \( 1 + 1.58T + 23T^{2} \) |
| 29 | \( 1 + 6.31T + 29T^{2} \) |
| 31 | \( 1 - 9.25T + 31T^{2} \) |
| 37 | \( 1 - 3.59T + 37T^{2} \) |
| 41 | \( 1 - 5.58T + 41T^{2} \) |
| 43 | \( 1 - 8.15T + 43T^{2} \) |
| 47 | \( 1 - 2.64T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 5.28T + 59T^{2} \) |
| 61 | \( 1 - 9.37T + 61T^{2} \) |
| 67 | \( 1 + 0.944T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 5.11T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 6.69T + 97T^{2} \) |
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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
−8.173326853412688077067173929656, −7.72989894661125196452513633033, −6.81236793476889465891390357534, −6.38818437880417513615752244694, −5.73116879861804978370793951163, −3.98794682392438037838556127654, −3.05427443119785764093284157218, −2.43122688203399923423101392237, −2.06169202475223398595070754577, −0.70833296136711141251812042853,
0.70833296136711141251812042853, 2.06169202475223398595070754577, 2.43122688203399923423101392237, 3.05427443119785764093284157218, 3.98794682392438037838556127654, 5.73116879861804978370793951163, 6.38818437880417513615752244694, 6.81236793476889465891390357534, 7.72989894661125196452513633033, 8.173326853412688077067173929656
