| L(s) = 1 | + 3.25·3-s + 7.62·9-s + 11-s + 5.29·13-s + 2.03·17-s − 4.48·19-s − 7.62·23-s + 15.0·27-s − 5.62·29-s − 1.22·31-s + 3.25·33-s + 7.62·37-s + 17.2·39-s + 9.77·41-s + 11.6·43-s + 6.51·47-s + 6.62·51-s − 11.2·53-s − 14.6·57-s − 5.29·59-s + 7.74·61-s + 8.24·67-s − 24.8·69-s + 2.37·71-s − 16.2·73-s + 5.62·79-s + 26.2·81-s + ⋯ |
| L(s) = 1 | + 1.88·3-s + 2.54·9-s + 0.301·11-s + 1.46·13-s + 0.492·17-s − 1.02·19-s − 1.58·23-s + 2.90·27-s − 1.04·29-s − 0.220·31-s + 0.567·33-s + 1.25·37-s + 2.76·39-s + 1.52·41-s + 1.77·43-s + 0.950·47-s + 0.927·51-s − 1.54·53-s − 1.93·57-s − 0.688·59-s + 0.991·61-s + 1.00·67-s − 2.99·69-s + 0.282·71-s − 1.90·73-s + 0.632·79-s + 2.91·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.193331832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.193331832\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 3.25T + 3T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 - 5.29T + 13T^{2} \) |
| 17 | \( 1 - 2.03T + 17T^{2} \) |
| 19 | \( 1 + 4.48T + 19T^{2} \) |
| 23 | \( 1 + 7.62T + 23T^{2} \) |
| 29 | \( 1 + 5.62T + 29T^{2} \) |
| 31 | \( 1 + 1.22T + 31T^{2} \) |
| 37 | \( 1 - 7.62T + 37T^{2} \) |
| 41 | \( 1 - 9.77T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 - 6.51T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 5.29T + 59T^{2} \) |
| 61 | \( 1 - 7.74T + 61T^{2} \) |
| 67 | \( 1 - 8.24T + 67T^{2} \) |
| 71 | \( 1 - 2.37T + 71T^{2} \) |
| 73 | \( 1 + 16.2T + 73T^{2} \) |
| 79 | \( 1 - 5.62T + 79T^{2} \) |
| 83 | \( 1 + 0.804T + 83T^{2} \) |
| 89 | \( 1 + 0.804T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−7.76472174393364188062449864010, −7.36856746849305484390296294879, −6.23685127960949394931745202134, −5.91862548876130480250317111688, −4.45853351333583315919584417871, −3.96510893904991235973868230525, −3.52597137777758898646553794001, −2.54354510713419792098326882296, −1.95430070246354500241172629491, −1.04849996911517172126367969196,
1.04849996911517172126367969196, 1.95430070246354500241172629491, 2.54354510713419792098326882296, 3.52597137777758898646553794001, 3.96510893904991235973868230525, 4.45853351333583315919584417871, 5.91862548876130480250317111688, 6.23685127960949394931745202134, 7.36856746849305484390296294879, 7.76472174393364188062449864010
