| L(s) = 1 | + 3·7-s + 3·11-s − 13-s + 7·17-s + 2·19-s + 23-s + 6·29-s + 10·31-s − 37-s − 41-s + 4·43-s − 4·47-s + 2·49-s − 53-s + 13·61-s + 8·67-s − 5·71-s − 10·73-s + 9·77-s + 79-s − 6·83-s + 9·89-s − 3·91-s + 97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.13·7-s + 0.904·11-s − 0.277·13-s + 1.69·17-s + 0.458·19-s + 0.208·23-s + 1.11·29-s + 1.79·31-s − 0.164·37-s − 0.156·41-s + 0.609·43-s − 0.583·47-s + 2/7·49-s − 0.137·53-s + 1.66·61-s + 0.977·67-s − 0.593·71-s − 1.17·73-s + 1.02·77-s + 0.112·79-s − 0.658·83-s + 0.953·89-s − 0.314·91-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.550160093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.550160093\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−13.99671506700471, −13.45845413372737, −12.77024606028922, −12.14161449350096, −11.84516198726632, −11.58575215380380, −10.88182241740984, −10.33235421129120, −9.818929949941532, −9.511176524723846, −8.663018320361577, −8.274164873027014, −7.915515290397468, −7.266600632278887, −6.778385266218654, −6.146407954730596, −5.556445464848389, −5.008558182958128, −4.552224802071002, −3.954816531808528, −3.225944965742727, −2.722731830997674, −1.855588161286254, −1.169761185266133, −0.7998492809242169,
0.7998492809242169, 1.169761185266133, 1.855588161286254, 2.722731830997674, 3.225944965742727, 3.954816531808528, 4.552224802071002, 5.008558182958128, 5.556445464848389, 6.146407954730596, 6.778385266218654, 7.266600632278887, 7.915515290397468, 8.274164873027014, 8.663018320361577, 9.511176524723846, 9.818929949941532, 10.33235421129120, 10.88182241740984, 11.58575215380380, 11.84516198726632, 12.14161449350096, 12.77024606028922, 13.45845413372737, 13.99671506700471
