| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 12-s + 13-s − 15-s + 16-s + 2·17-s − 18-s + 4·19-s + 20-s + 2·23-s + 24-s + 25-s − 26-s − 27-s + 6·29-s + 30-s − 32-s − 2·34-s + 36-s + 2·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.417·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 1.11·29-s + 0.182·30-s − 0.176·32-s − 0.342·34-s + 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.04614351958739, −14.31355194905168, −13.68163147694453, −13.47534851994519, −12.56324768827032, −12.18136462254052, −11.76626244351340, −11.11981270039521, −10.57903358160374, −10.27119017857736, −9.590363061729302, −9.213727943420524, −8.619014705125723, −7.909355191568750, −7.512672341743437, −6.768478824713615, −6.400691589258106, −5.714755934336287, −5.248616025635366, −4.611117705544412, −3.775978940314519, −3.025748346856269, −2.455432522417944, −1.396015163492792, −1.083129683694531, 0,
1.083129683694531, 1.396015163492792, 2.455432522417944, 3.025748346856269, 3.775978940314519, 4.611117705544412, 5.248616025635366, 5.714755934336287, 6.400691589258106, 6.768478824713615, 7.512672341743437, 7.909355191568750, 8.619014705125723, 9.213727943420524, 9.590363061729302, 10.27119017857736, 10.57903358160374, 11.11981270039521, 11.76626244351340, 12.18136462254052, 12.56324768827032, 13.47534851994519, 13.68163147694453, 14.31355194905168, 15.04614351958739
