| L(s) = 1 | − 2.23·2-s + 3-s + 2.97·4-s − 0.893·5-s − 2.23·6-s − 2.18·8-s + 9-s + 1.99·10-s − 3.03·11-s + 2.97·12-s − 1.02·13-s − 0.893·15-s − 1.08·16-s − 0.590·17-s − 2.23·18-s − 3.50·19-s − 2.65·20-s + 6.76·22-s − 0.556·23-s − 2.18·24-s − 4.20·25-s + 2.28·26-s + 27-s + 7.89·29-s + 1.99·30-s + 1.28·31-s + 6.79·32-s + ⋯ |
| L(s) = 1 | − 1.57·2-s + 0.577·3-s + 1.48·4-s − 0.399·5-s − 0.910·6-s − 0.771·8-s + 0.333·9-s + 0.630·10-s − 0.914·11-s + 0.859·12-s − 0.283·13-s − 0.230·15-s − 0.271·16-s − 0.143·17-s − 0.525·18-s − 0.804·19-s − 0.594·20-s + 1.44·22-s − 0.115·23-s − 0.445·24-s − 0.840·25-s + 0.447·26-s + 0.192·27-s + 1.46·29-s + 0.363·30-s + 0.230·31-s + 1.20·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6051235739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6051235739\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 5 | \( 1 + 0.893T + 5T^{2} \) |
| 11 | \( 1 + 3.03T + 11T^{2} \) |
| 13 | \( 1 + 1.02T + 13T^{2} \) |
| 17 | \( 1 + 0.590T + 17T^{2} \) |
| 19 | \( 1 + 3.50T + 19T^{2} \) |
| 23 | \( 1 + 0.556T + 23T^{2} \) |
| 29 | \( 1 - 7.89T + 29T^{2} \) |
| 31 | \( 1 - 1.28T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 43 | \( 1 + 5.76T + 43T^{2} \) |
| 47 | \( 1 - 0.229T + 47T^{2} \) |
| 53 | \( 1 - 8.54T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 8.09T + 61T^{2} \) |
| 67 | \( 1 - 8.13T + 67T^{2} \) |
| 71 | \( 1 - 1.38T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 4.67T + 79T^{2} \) |
| 83 | \( 1 - 3.22T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 - 5.16T + 97T^{2} \) |
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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
−8.102724421406290057384581623835, −7.77003466887083646812306209009, −6.94399492469264022381223161775, −6.38876923904216168066650506039, −5.17862299948950945590114925318, −4.39826629692497276432221219142, −3.36713806089681207927445296299, −2.44328963937335634681052745926, −1.77832969501879149481219902759, −0.48973628615642097886555838330,
0.48973628615642097886555838330, 1.77832969501879149481219902759, 2.44328963937335634681052745926, 3.36713806089681207927445296299, 4.39826629692497276432221219142, 5.17862299948950945590114925318, 6.38876923904216168066650506039, 6.94399492469264022381223161775, 7.77003466887083646812306209009, 8.102724421406290057384581623835
