| L(s) = 1 | − 0.772·2-s − 1.40·4-s + 2.30·5-s + 3.13·7-s + 2.62·8-s − 1.78·10-s + 0.310·11-s + 4.97·13-s − 2.41·14-s + 0.778·16-s − 3.86·17-s + 19-s − 3.24·20-s − 0.239·22-s − 6.16·23-s + 0.330·25-s − 3.84·26-s − 4.39·28-s − 7.15·29-s + 3.98·31-s − 5.85·32-s + 2.98·34-s + 7.23·35-s + 2.52·37-s − 0.772·38-s + 6.06·40-s + 9.83·41-s + ⋯ |
| L(s) = 1 | − 0.545·2-s − 0.701·4-s + 1.03·5-s + 1.18·7-s + 0.929·8-s − 0.563·10-s + 0.0936·11-s + 1.37·13-s − 0.646·14-s + 0.194·16-s − 0.937·17-s + 0.229·19-s − 0.724·20-s − 0.0511·22-s − 1.28·23-s + 0.0660·25-s − 0.753·26-s − 0.831·28-s − 1.32·29-s + 0.715·31-s − 1.03·32-s + 0.511·34-s + 1.22·35-s + 0.415·37-s − 0.125·38-s + 0.959·40-s + 1.53·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.963432781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.963432781\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 0.772T + 2T^{2} \) |
| 5 | \( 1 - 2.30T + 5T^{2} \) |
| 7 | \( 1 - 3.13T + 7T^{2} \) |
| 11 | \( 1 - 0.310T + 11T^{2} \) |
| 13 | \( 1 - 4.97T + 13T^{2} \) |
| 17 | \( 1 + 3.86T + 17T^{2} \) |
| 23 | \( 1 + 6.16T + 23T^{2} \) |
| 29 | \( 1 + 7.15T + 29T^{2} \) |
| 31 | \( 1 - 3.98T + 31T^{2} \) |
| 37 | \( 1 - 2.52T + 37T^{2} \) |
| 41 | \( 1 - 9.83T + 41T^{2} \) |
| 43 | \( 1 + 0.806T + 43T^{2} \) |
| 53 | \( 1 - 1.15T + 53T^{2} \) |
| 59 | \( 1 - 0.741T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 4.03T + 67T^{2} \) |
| 71 | \( 1 + 1.51T + 71T^{2} \) |
| 73 | \( 1 - 5.26T + 73T^{2} \) |
| 79 | \( 1 - 7.25T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 + 8.07T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−8.135023546781525280797295929528, −7.34036385055172403232835805940, −6.30866815777039746119836581609, −5.77068532175653720418424979765, −5.09271396590246511078503824329, −4.26466649313060945847285252296, −3.74900912834796206434354457334, −2.24921187475977750518051760233, −1.70053472746850978778783000658, −0.806763468295952102522059975825,
0.806763468295952102522059975825, 1.70053472746850978778783000658, 2.24921187475977750518051760233, 3.74900912834796206434354457334, 4.26466649313060945847285252296, 5.09271396590246511078503824329, 5.77068532175653720418424979765, 6.30866815777039746119836581609, 7.34036385055172403232835805940, 8.135023546781525280797295929528
