| L(s) = 1 | − 3-s − 4·7-s + 9-s + 11-s + 4·13-s + 17-s − 6·19-s + 4·21-s − 27-s + 8·29-s − 33-s + 6·37-s − 4·39-s − 8·41-s + 4·43-s + 8·47-s + 9·49-s − 51-s − 2·53-s + 6·57-s + 4·59-s − 2·61-s − 4·63-s − 4·67-s − 4·73-s − 4·77-s − 14·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 0.242·17-s − 1.37·19-s + 0.872·21-s − 0.192·27-s + 1.48·29-s − 0.174·33-s + 0.986·37-s − 0.640·39-s − 1.24·41-s + 0.609·43-s + 1.16·47-s + 9/7·49-s − 0.140·51-s − 0.274·53-s + 0.794·57-s + 0.520·59-s − 0.256·61-s − 0.503·63-s − 0.488·67-s − 0.468·73-s − 0.455·77-s − 1.57·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.359777342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.359777342\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.82291768670291, −12.56455927587605, −12.09323975811948, −11.58815347492033, −11.06800455277034, −10.54513783602383, −10.21014738484537, −9.834045246113494, −9.156645706239815, −8.788840344387875, −8.356105333661316, −7.709175743037947, −7.014675040490031, −6.627500719054276, −6.239262257398883, −5.933010194360711, −5.367291594759625, −4.513602627932045, −4.177345519625302, −3.640236526832171, −3.022219201668817, −2.560125418098465, −1.706637953402092, −0.9963578542590281, −0.3941865534694046,
0.3941865534694046, 0.9963578542590281, 1.706637953402092, 2.560125418098465, 3.022219201668817, 3.640236526832171, 4.177345519625302, 4.513602627932045, 5.367291594759625, 5.933010194360711, 6.239262257398883, 6.627500719054276, 7.014675040490031, 7.709175743037947, 8.356105333661316, 8.788840344387875, 9.156645706239815, 9.834045246113494, 10.21014738484537, 10.54513783602383, 11.06800455277034, 11.58815347492033, 12.09323975811948, 12.56455927587605, 12.82291768670291
