| L(s) = 1 | + 3-s − 2·7-s + 9-s + 2·11-s + 2·13-s + 4·17-s + 6·19-s − 2·21-s + 6·23-s + 27-s − 10·29-s + 8·31-s + 2·33-s − 2·37-s + 2·39-s + 2·41-s − 43-s − 2·47-s − 3·49-s + 4·51-s − 10·53-s + 6·57-s − 2·59-s − 12·61-s − 2·63-s − 12·67-s + 6·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s + 0.970·17-s + 1.37·19-s − 0.436·21-s + 1.25·23-s + 0.192·27-s − 1.85·29-s + 1.43·31-s + 0.348·33-s − 0.328·37-s + 0.320·39-s + 0.312·41-s − 0.152·43-s − 0.291·47-s − 3/7·49-s + 0.560·51-s − 1.37·53-s + 0.794·57-s − 0.260·59-s − 1.53·61-s − 0.251·63-s − 1.46·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.59555802222109, −14.37431299421198, −13.50170413195924, −13.40982680811297, −12.83315898859150, −12.10153981045566, −11.83026853783795, −11.11809735852547, −10.61409898094715, −9.866551681694962, −9.546336056833197, −9.104905003453585, −8.585408998556959, −7.821596636952734, −7.394557407789476, −6.939456424798128, −6.088274323145786, −5.847538518755997, −4.965443736295002, −4.395424660462435, −3.533211688584711, −3.198518604310133, −2.771578610288841, −1.465270225622881, −1.269275253717526, 0,
1.269275253717526, 1.465270225622881, 2.771578610288841, 3.198518604310133, 3.533211688584711, 4.395424660462435, 4.965443736295002, 5.847538518755997, 6.088274323145786, 6.939456424798128, 7.394557407789476, 7.821596636952734, 8.585408998556959, 9.104905003453585, 9.546336056833197, 9.866551681694962, 10.61409898094715, 11.11809735852547, 11.83026853783795, 12.10153981045566, 12.83315898859150, 13.40982680811297, 13.50170413195924, 14.37431299421198, 14.59555802222109
