| L(s) = 1 | + 2·3-s + 7-s + 9-s − 4·11-s + 2·13-s + 4·17-s + 2·21-s − 23-s − 4·27-s + 2·29-s + 6·31-s − 8·33-s − 2·37-s + 4·39-s − 6·41-s − 4·43-s − 10·47-s + 49-s + 8·51-s − 6·53-s + 2·59-s + 8·61-s + 63-s − 4·67-s − 2·69-s + 16·71-s − 6·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.970·17-s + 0.436·21-s − 0.208·23-s − 0.769·27-s + 0.371·29-s + 1.07·31-s − 1.39·33-s − 0.328·37-s + 0.640·39-s − 0.937·41-s − 0.609·43-s − 1.45·47-s + 1/7·49-s + 1.12·51-s − 0.824·53-s + 0.260·59-s + 1.02·61-s + 0.125·63-s − 0.488·67-s − 0.240·69-s + 1.89·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.437447225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.437447225\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.12911919152989, −13.76385356065400, −13.43411945439688, −12.80600393977555, −12.37193041758048, −11.59901296096707, −11.31030394832802, −10.53959478294141, −10.01489605263765, −9.779279969093481, −8.960161962224581, −8.390818980716094, −8.151790764017498, −7.762928694626021, −7.097731361001585, −6.394391107292582, −5.795463023075170, −5.074388664017233, −4.755734796848261, −3.771048517513299, −3.327099566244907, −2.833276706333149, −2.136302935805551, −1.541509322444067, −0.5659857048946671,
0.5659857048946671, 1.541509322444067, 2.136302935805551, 2.833276706333149, 3.327099566244907, 3.771048517513299, 4.755734796848261, 5.074388664017233, 5.795463023075170, 6.394391107292582, 7.097731361001585, 7.762928694626021, 8.151790764017498, 8.390818980716094, 8.960161962224581, 9.779279969093481, 10.01489605263765, 10.53959478294141, 11.31030394832802, 11.59901296096707, 12.37193041758048, 12.80600393977555, 13.43411945439688, 13.76385356065400, 14.12911919152989
