| L(s) = 1 | + 2·3-s − 7-s + 9-s + 4·13-s − 6·17-s + 2·19-s − 2·21-s − 5·25-s − 4·27-s + 6·29-s − 4·31-s + 2·37-s + 8·39-s + 6·41-s + 8·43-s − 12·47-s + 49-s − 12·51-s + 6·53-s + 4·57-s + 6·59-s + 8·61-s − 63-s − 4·67-s + 2·73-s − 10·75-s − 8·79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.10·13-s − 1.45·17-s + 0.458·19-s − 0.436·21-s − 25-s − 0.769·27-s + 1.11·29-s − 0.718·31-s + 0.328·37-s + 1.28·39-s + 0.937·41-s + 1.21·43-s − 1.75·47-s + 1/7·49-s − 1.68·51-s + 0.824·53-s + 0.529·57-s + 0.781·59-s + 1.02·61-s − 0.125·63-s − 0.488·67-s + 0.234·73-s − 1.15·75-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.126928339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.126928339\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.12917195101859, −12.73659383495597, −11.87104041272984, −11.35204021519883, −11.28480321790200, −10.43117831481402, −10.09697083181932, −9.493692497204458, −9.004114579819869, −8.791989974446936, −8.317123806205391, −7.718176324562899, −7.432752371940186, −6.627372272243193, −6.345287771861489, −5.773697051546522, −5.200276543673204, −4.455770860890105, −3.868509578320793, −3.704799121705127, −2.877484887540865, −2.531090506649586, −1.951268525630507, −1.268660890805920, −0.4442676835914295,
0.4442676835914295, 1.268660890805920, 1.951268525630507, 2.531090506649586, 2.877484887540865, 3.704799121705127, 3.868509578320793, 4.455770860890105, 5.200276543673204, 5.773697051546522, 6.345287771861489, 6.627372272243193, 7.432752371940186, 7.718176324562899, 8.317123806205391, 8.791989974446936, 9.004114579819869, 9.493692497204458, 10.09697083181932, 10.43117831481402, 11.28480321790200, 11.35204021519883, 11.87104041272984, 12.73659383495597, 13.12917195101859
