| L(s) = 1 | − 3-s + 9-s + 4·11-s + 2·13-s − 2·17-s − 4·19-s − 27-s − 2·29-s − 4·33-s + 10·37-s − 2·39-s + 10·41-s + 4·43-s + 8·47-s − 7·49-s + 2·51-s + 10·53-s + 4·57-s + 4·59-s − 2·61-s + 12·67-s + 8·71-s − 10·73-s + 81-s + 12·83-s + 2·87-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s − 0.192·27-s − 0.371·29-s − 0.696·33-s + 1.64·37-s − 0.320·39-s + 1.56·41-s + 0.609·43-s + 1.16·47-s − 49-s + 0.280·51-s + 1.37·53-s + 0.529·57-s + 0.520·59-s − 0.256·61-s + 1.46·67-s + 0.949·71-s − 1.17·73-s + 1/9·81-s + 1.31·83-s + 0.214·87-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.427722446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.427722446\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−9.655893575783761174602693945437, −9.049118048765622993735733025092, −8.151425844540532890281788534319, −7.10478788988449333222322211892, −6.34199635143511529870167101224, −5.72078508715635747921690343280, −4.42801570520906772580093162574, −3.86223113294681320091712334301, −2.32158421366848578869138486097, −0.957897388351054006923114027189,
0.957897388351054006923114027189, 2.32158421366848578869138486097, 3.86223113294681320091712334301, 4.42801570520906772580093162574, 5.72078508715635747921690343280, 6.34199635143511529870167101224, 7.10478788988449333222322211892, 8.151425844540532890281788534319, 9.049118048765622993735733025092, 9.655893575783761174602693945437
