| L(s) = 1 | + 3-s − 5-s + 9-s − 4·11-s + 2·13-s − 15-s + 2·17-s + 19-s − 4·23-s + 25-s + 27-s + 6·29-s − 4·31-s − 4·33-s − 6·37-s + 2·39-s + 10·41-s + 4·43-s − 45-s + 12·47-s − 7·49-s + 2·51-s + 6·53-s + 4·55-s + 57-s + 12·59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.258·15-s + 0.485·17-s + 0.229·19-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.718·31-s − 0.696·33-s − 0.986·37-s + 0.320·39-s + 1.56·41-s + 0.609·43-s − 0.149·45-s + 1.75·47-s − 49-s + 0.280·51-s + 0.824·53-s + 0.539·55-s + 0.132·57-s + 1.56·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.023796990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.023796990\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−8.283105266901293775359286481226, −7.65793136383988639647889878637, −7.15617417235089279548403004517, −6.07061566169635625281315866590, −5.40891625710435688403964718382, −4.48026541587866654317061795722, −3.71741657293176102857512785058, −2.92108927585535703482944330075, −2.09210044862115434120313168487, −0.76461290581560603486592018258,
0.76461290581560603486592018258, 2.09210044862115434120313168487, 2.92108927585535703482944330075, 3.71741657293176102857512785058, 4.48026541587866654317061795722, 5.40891625710435688403964718382, 6.07061566169635625281315866590, 7.15617417235089279548403004517, 7.65793136383988639647889878637, 8.283105266901293775359286481226
