L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 2·7-s − 8-s + 9-s + 10-s − 11-s + 12-s + 4·13-s + 2·14-s − 15-s + 16-s + 17-s − 18-s + 5·19-s − 20-s − 2·21-s + 22-s − 4·23-s − 24-s + 25-s − 4·26-s + 27-s − 2·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s + 0.288·12-s + 1.10·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.14·19-s − 0.223·20-s − 0.436·21-s + 0.213·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 43 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.69911363856733, −14.15616058092493, −13.66092188926665, −13.11038055955506, −12.63424391956441, −12.10670117372293, −11.51402882711722, −10.98014672874003, −10.52556109355233, −9.885472211285627, −9.421005160110048, −9.061210631912894, −8.355144008311095, −7.946691066976729, −7.503686816505738, −6.808458436595420, −6.433055542916403, −5.576605394943334, −5.217609064884577, −4.051530586427270, −3.632067023904768, −3.202828187558438, −2.424599507736841, −1.669633601327868, −0.9077276484859419, 0,
0.9077276484859419, 1.669633601327868, 2.424599507736841, 3.202828187558438, 3.632067023904768, 4.051530586427270, 5.217609064884577, 5.576605394943334, 6.433055542916403, 6.808458436595420, 7.503686816505738, 7.946691066976729, 8.355144008311095, 9.061210631912894, 9.421005160110048, 9.885472211285627, 10.52556109355233, 10.98014672874003, 11.51402882711722, 12.10670117372293, 12.63424391956441, 13.11038055955506, 13.66092188926665, 14.15616058092493, 14.69911363856733