| L(s) = 1 | + 3-s − 2·4-s + 7-s − 2·9-s − 11-s − 2·12-s + 2·13-s + 4·16-s + 6·17-s − 7·19-s + 21-s − 3·23-s − 5·25-s − 5·27-s − 2·28-s + 29-s − 4·31-s − 33-s + 4·36-s + 2·37-s + 2·39-s + 9·41-s − 4·43-s + 2·44-s − 9·47-s + 4·48-s + 49-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.377·7-s − 2/3·9-s − 0.301·11-s − 0.577·12-s + 0.554·13-s + 16-s + 1.45·17-s − 1.60·19-s + 0.218·21-s − 0.625·23-s − 25-s − 0.962·27-s − 0.377·28-s + 0.185·29-s − 0.718·31-s − 0.174·33-s + 2/3·36-s + 0.328·37-s + 0.320·39-s + 1.40·41-s − 0.609·43-s + 0.301·44-s − 1.31·47-s + 0.577·48-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2233 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2233 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.449426738704921033911123355724, −8.211465775907911429013068789934, −7.42532714704336028183876296579, −5.95843087654965787151006295911, −5.62452236100427465906973804385, −4.44220881558004033390294560419, −3.78501230616501816766088607371, −2.84410792856064290796363321251, −1.59581542715941783964660566261, 0,
1.59581542715941783964660566261, 2.84410792856064290796363321251, 3.78501230616501816766088607371, 4.44220881558004033390294560419, 5.62452236100427465906973804385, 5.95843087654965787151006295911, 7.42532714704336028183876296579, 8.211465775907911429013068789934, 8.449426738704921033911123355724
