| L(s) = 1 | + 18.4·5-s − 20.1·7-s + 81·9-s + 59.8·11-s + 572.·17-s + 361·19-s − 158·23-s − 283.·25-s − 371.·35-s − 2.72e3·43-s + 1.49e3·45-s − 4.28e3·47-s − 1.99e3·49-s + 1.10e3·55-s + 4.24e3·61-s − 1.63e3·63-s + 8.13e3·73-s − 1.20e3·77-s + 6.56e3·81-s − 5.67e3·83-s + 1.05e4·85-s + 6.66e3·95-s + 4.84e3·99-s − 9.99e3·101-s − 2.91e3·115-s − 1.15e4·119-s + ⋯ |
| L(s) = 1 | + 0.738·5-s − 0.410·7-s + 9-s + 0.494·11-s + 1.98·17-s + 19-s − 0.298·23-s − 0.453·25-s − 0.303·35-s − 1.47·43-s + 0.738·45-s − 1.93·47-s − 0.831·49-s + 0.365·55-s + 1.14·61-s − 0.410·63-s + 1.52·73-s − 0.203·77-s + 81-s − 0.824·83-s + 1.46·85-s + 0.738·95-s + 0.494·99-s − 0.980·101-s − 0.220·115-s − 0.814·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.970925934\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.970925934\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 361T \) |
| good | 3 | \( 1 - 81T^{2} \) |
| 5 | \( 1 - 18.4T + 625T^{2} \) |
| 7 | \( 1 + 20.1T + 2.40e3T^{2} \) |
| 11 | \( 1 - 59.8T + 1.46e4T^{2} \) |
| 13 | \( 1 - 2.85e4T^{2} \) |
| 17 | \( 1 - 572.T + 8.35e4T^{2} \) |
| 23 | \( 1 + 158T + 2.79e5T^{2} \) |
| 29 | \( 1 - 7.07e5T^{2} \) |
| 31 | \( 1 - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.82e6T^{2} \) |
| 43 | \( 1 + 2.72e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 4.28e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.21e7T^{2} \) |
| 61 | \( 1 - 4.24e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 2.01e7T^{2} \) |
| 71 | \( 1 - 2.54e7T^{2} \) |
| 73 | \( 1 - 8.13e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.89e7T^{2} \) |
| 83 | \( 1 + 5.67e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 6.27e7T^{2} \) |
| 97 | \( 1 - 8.85e7T^{2} \) |
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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
−13.74744562895246542081352818785, −12.74095326069789419674494380532, −11.70864465853710102278416160415, −9.981556671109526718291112115944, −9.682769044989625067952066346482, −7.894755410937480218167634326307, −6.60156746884170893242163577892, −5.29732262375819872334377315942, −3.49104535304896258821772151362, −1.42783578818619358528327281108,
1.42783578818619358528327281108, 3.49104535304896258821772151362, 5.29732262375819872334377315942, 6.60156746884170893242163577892, 7.894755410937480218167634326307, 9.682769044989625067952066346482, 9.981556671109526718291112115944, 11.70864465853710102278416160415, 12.74095326069789419674494380532, 13.74744562895246542081352818785
