L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 2·7-s − 8-s + 9-s + 10-s + 4·11-s − 12-s − 2·14-s + 15-s + 16-s + 17-s − 18-s − 20-s − 2·21-s − 4·22-s + 8·23-s + 24-s + 25-s − 27-s + 2·28-s − 8·29-s − 30-s + 8·31-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 + 1.20·11-s − 0.288·12-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.223·20-s − 0.436·21-s − 0.852·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.377·28-s − 1.48·29-s − 0.182·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−14.36745839551061, −13.63541207212651, −13.07728493845405, −12.51684489136846, −11.98849643459839, −11.55333390179349, −11.16326197669960, −10.93308618305798, −10.09443094821537, −9.745356402337775, −9.003809049591944, −8.783326271835773, −8.080674183277893, −7.588976115879501, −7.065437691966549, −6.626513617966400, −6.048949207046494, −5.411222067622575, −4.760328651533251, −4.357647336659294, −3.540048135739730, −3.049963425297531, −2.081960321094442, −1.364253295131277, −0.9820015724674870, 0,
0.9820015724674870, 1.364253295131277, 2.081960321094442, 3.049963425297531, 3.540048135739730, 4.357647336659294, 4.760328651533251, 5.411222067622575, 6.048949207046494, 6.626513617966400, 7.065437691966549, 7.588976115879501, 8.080674183277893, 8.783326271835773, 9.003809049591944, 9.745356402337775, 10.09443094821537, 10.93308618305798, 11.16326197669960, 11.55333390179349, 11.98849643459839, 12.51684489136846, 13.07728493845405, 13.63541207212651, 14.36745839551061