| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 3·11-s − 12-s + 13-s + 14-s + 15-s + 16-s + 6·17-s − 18-s + 3·19-s − 20-s + 21-s − 3·22-s + 2·23-s + 24-s + 25-s − 26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.904·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.688·19-s − 0.223·20-s + 0.218·21-s − 0.639·22-s + 0.417·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−12.59730912806401, −12.07603243939673, −11.72651340222784, −11.40018705635892, −10.99179600563509, −10.35146550317453, −10.05041769154384, −9.470396321022549, −9.246339746063391, −8.687372568802587, −8.088089927425658, −7.572474673236048, −7.384021374301587, −6.745857133164723, −6.275631715427098, −5.816642732028429, −5.470114195657244, −4.654266150722255, −4.294785189686910, −3.517482096901953, −3.219221143891126, −2.691073050833925, −1.611544056254203, −1.348615185874191, −0.7292834222971563, 0,
0.7292834222971563, 1.348615185874191, 1.611544056254203, 2.691073050833925, 3.219221143891126, 3.517482096901953, 4.294785189686910, 4.654266150722255, 5.470114195657244, 5.816642732028429, 6.275631715427098, 6.745857133164723, 7.384021374301587, 7.572474673236048, 8.088089927425658, 8.687372568802587, 9.246339746063391, 9.470396321022549, 10.05041769154384, 10.35146550317453, 10.99179600563509, 11.40018705635892, 11.72651340222784, 12.07603243939673, 12.59730912806401
