| L(s) = 1 | − 2-s − 4-s − 2·5-s + 3·8-s + 2·10-s − 4·11-s − 2·13-s − 16-s − 6·17-s + 2·20-s + 4·22-s − 25-s + 2·26-s − 2·29-s − 5·32-s + 6·34-s − 6·37-s − 6·40-s − 2·41-s − 4·43-s + 4·44-s + 50-s + 2·52-s + 6·53-s + 8·55-s + 2·58-s − 12·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.894·5-s + 1.06·8-s + 0.632·10-s − 1.20·11-s − 0.554·13-s − 1/4·16-s − 1.45·17-s + 0.447·20-s + 0.852·22-s − 1/5·25-s + 0.392·26-s − 0.371·29-s − 0.883·32-s + 1.02·34-s − 0.986·37-s − 0.948·40-s − 0.312·41-s − 0.609·43-s + 0.603·44-s + 0.141·50-s + 0.277·52-s + 0.824·53-s + 1.07·55-s + 0.262·58-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.58479926748091, −13.32956705146113, −12.85196651487021, −12.33256530370328, −11.79895401413914, −11.28265369958407, −10.81967205866097, −10.34222096137701, −10.04619574278562, −9.312028814032149, −8.935336808339231, −8.468144798181291, −8.014654432941583, −7.487715485647086, −7.291883505892065, −6.563914002998695, −5.914206367633884, −5.107716161782147, −4.836878954265275, −4.376586694003391, −3.655949753886024, −3.244902852346174, −2.230109395067092, −1.989583292537695, −0.9027801776158123, 0, 0,
0.9027801776158123, 1.989583292537695, 2.230109395067092, 3.244902852346174, 3.655949753886024, 4.376586694003391, 4.836878954265275, 5.107716161782147, 5.914206367633884, 6.563914002998695, 7.291883505892065, 7.487715485647086, 8.014654432941583, 8.468144798181291, 8.935336808339231, 9.312028814032149, 10.04619574278562, 10.34222096137701, 10.81967205866097, 11.28265369958407, 11.79895401413914, 12.33256530370328, 12.85196651487021, 13.32956705146113, 13.58479926748091
