| L(s) = 1 | + 4·5-s − 7-s − 2·11-s + 13-s − 4·17-s + 3·19-s + 11·25-s − 4·29-s − 4·35-s + 2·37-s + 5·43-s − 4·47-s − 6·49-s − 6·53-s − 8·55-s + 10·59-s + 2·61-s + 4·65-s + 5·67-s − 10·71-s + 9·73-s + 2·77-s − 9·79-s − 16·85-s − 10·89-s − 91-s + 12·95-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.377·7-s − 0.603·11-s + 0.277·13-s − 0.970·17-s + 0.688·19-s + 11/5·25-s − 0.742·29-s − 0.676·35-s + 0.328·37-s + 0.762·43-s − 0.583·47-s − 6/7·49-s − 0.824·53-s − 1.07·55-s + 1.30·59-s + 0.256·61-s + 0.496·65-s + 0.610·67-s − 1.18·71-s + 1.05·73-s + 0.227·77-s − 1.01·79-s − 1.73·85-s − 1.05·89-s − 0.104·91-s + 1.23·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.132674023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.132674023\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−13.50162557810982, −13.22361526215514, −12.79621154559206, −12.43815651478628, −11.48842969267253, −11.13998003691438, −10.69100406016438, −9.952152111129630, −9.803131203628926, −9.369970613315118, −8.693486686422461, −8.425453082594761, −7.480473695309339, −7.133321600068649, −6.349752042924289, −6.167096358215799, −5.544432041718806, −5.099707043767785, −4.565514222637309, −3.739912547752562, −3.043038661889346, −2.523761020586368, −1.973643915927954, −1.402497241904401, −0.5266259800903396,
0.5266259800903396, 1.402497241904401, 1.973643915927954, 2.523761020586368, 3.043038661889346, 3.739912547752562, 4.565514222637309, 5.099707043767785, 5.544432041718806, 6.167096358215799, 6.349752042924289, 7.133321600068649, 7.480473695309339, 8.425453082594761, 8.693486686422461, 9.369970613315118, 9.803131203628926, 9.952152111129630, 10.69100406016438, 11.13998003691438, 11.48842969267253, 12.43815651478628, 12.79621154559206, 13.22361526215514, 13.50162557810982
