| L(s) = 1 | − 3-s − 4·7-s + 9-s − 11-s − 6·13-s + 17-s + 6·19-s + 4·21-s − 5·25-s − 27-s + 2·29-s − 8·31-s + 33-s − 2·37-s + 6·39-s − 2·41-s + 2·43-s + 6·47-s + 9·49-s − 51-s + 2·53-s − 6·57-s + 2·59-s − 8·61-s − 4·63-s − 8·67-s + 4·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 0.242·17-s + 1.37·19-s + 0.872·21-s − 25-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.174·33-s − 0.328·37-s + 0.960·39-s − 0.312·41-s + 0.304·43-s + 0.875·47-s + 9/7·49-s − 0.140·51-s + 0.274·53-s − 0.794·57-s + 0.260·59-s − 1.02·61-s − 0.503·63-s − 0.977·67-s + 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 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
−15.32739147776178, −14.70285304047701, −14.03489708054528, −13.56221250869498, −13.01512532938610, −12.44051446101876, −12.07897784966909, −11.74533000577615, −10.80799355163420, −10.38937271043472, −9.804171715195956, −9.418289187420355, −9.101488746777828, −7.876947006469639, −7.582709352257364, −6.950076625274811, −6.536793555883307, −5.619931980146919, −5.459825015567579, −4.704404249536385, −3.864882283421102, −3.290706978293556, −2.646595788135846, −1.889823492007914, −0.7023501468233069, 0,
0.7023501468233069, 1.889823492007914, 2.646595788135846, 3.290706978293556, 3.864882283421102, 4.704404249536385, 5.459825015567579, 5.619931980146919, 6.536793555883307, 6.950076625274811, 7.582709352257364, 7.876947006469639, 9.101488746777828, 9.418289187420355, 9.804171715195956, 10.38937271043472, 10.80799355163420, 11.74533000577615, 12.07897784966909, 12.44051446101876, 13.01512532938610, 13.56221250869498, 14.03489708054528, 14.70285304047701, 15.32739147776178
