| L(s) = 1 | + 2-s − 3-s − 4-s + 2·5-s − 6-s − 3·8-s + 9-s + 2·10-s + 12-s − 2·13-s − 2·15-s − 16-s − 6·17-s + 18-s + 4·19-s − 2·20-s + 3·24-s − 25-s − 2·26-s − 27-s + 2·29-s − 2·30-s + 5·32-s − 6·34-s − 36-s + 6·37-s + 4·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s − 1.06·8-s + 1/3·9-s + 0.632·10-s + 0.288·12-s − 0.554·13-s − 0.516·15-s − 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s + 0.612·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.371·29-s − 0.365·30-s + 0.883·32-s − 1.02·34-s − 1/6·36-s + 0.986·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17787 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17787 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 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
−16.01227889858686, −15.44932324068807, −14.96307581485170, −14.26758915193183, −13.80024995409471, −13.39010654330217, −12.91105860350334, −12.30720862686405, −11.80584273975452, −11.17612132607184, −10.54251418328237, −9.838116541789092, −9.387956130797399, −8.981766941919894, −8.116871407954105, −7.369409496376228, −6.621057175524006, −6.048257885833171, −5.618837291130084, −4.880391790569412, −4.497906540604975, −3.736512787508444, −2.783826182891050, −2.173773837483659, −1.056423424427328, 0,
1.056423424427328, 2.173773837483659, 2.783826182891050, 3.736512787508444, 4.497906540604975, 4.880391790569412, 5.618837291130084, 6.048257885833171, 6.621057175524006, 7.369409496376228, 8.116871407954105, 8.981766941919894, 9.387956130797399, 9.838116541789092, 10.54251418328237, 11.17612132607184, 11.80584273975452, 12.30720862686405, 12.91105860350334, 13.39010654330217, 13.80024995409471, 14.26758915193183, 14.96307581485170, 15.44932324068807, 16.01227889858686
