| L(s) = 1 | − 3-s − 7-s + 9-s − 13-s + 7·17-s + 19-s + 21-s − 4·23-s − 27-s − 10·31-s + 37-s + 39-s + 7·41-s − 4·43-s + 8·47-s + 49-s − 7·51-s − 53-s − 57-s + 5·61-s − 63-s − 13·67-s + 4·69-s + 5·71-s + 13·73-s + 14·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.277·13-s + 1.69·17-s + 0.229·19-s + 0.218·21-s − 0.834·23-s − 0.192·27-s − 1.79·31-s + 0.164·37-s + 0.160·39-s + 1.09·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.980·51-s − 0.137·53-s − 0.132·57-s + 0.640·61-s − 0.125·63-s − 1.58·67-s + 0.481·69-s + 0.593·71-s + 1.52·73-s + 1.57·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.935665136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.935665136\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−12.83692766160342, −12.27627791499059, −11.97639678531198, −11.59731668194657, −10.82856068502938, −10.66007014938645, −9.995216714292839, −9.732929257346324, −9.168599067902761, −8.782574457821488, −7.912602763613868, −7.606462781975387, −7.347820183301652, −6.553557600124358, −6.135058812502774, −5.681607338361740, −5.212830168765927, −4.806403480146645, −3.894729679316095, −3.699752609757591, −3.082345896682816, −2.306374511290534, −1.791067884679495, −0.9717029166146749, −0.4660202830319780,
0.4660202830319780, 0.9717029166146749, 1.791067884679495, 2.306374511290534, 3.082345896682816, 3.699752609757591, 3.894729679316095, 4.806403480146645, 5.212830168765927, 5.681607338361740, 6.135058812502774, 6.553557600124358, 7.347820183301652, 7.606462781975387, 7.912602763613868, 8.782574457821488, 9.168599067902761, 9.732929257346324, 9.995216714292839, 10.66007014938645, 10.82856068502938, 11.59731668194657, 11.97639678531198, 12.27627791499059, 12.83692766160342
