| L(s) = 1 | − 2-s − 3-s − 4-s + 6-s + 7-s + 3·8-s − 2·9-s + 12-s − 2·13-s − 14-s − 16-s − 4·17-s + 2·18-s − 4·19-s − 21-s + 6·23-s − 3·24-s + 2·26-s + 5·27-s − 28-s + 9·29-s − 5·32-s + 4·34-s + 2·36-s − 3·37-s + 4·38-s + 2·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s + 0.377·7-s + 1.06·8-s − 2/3·9-s + 0.288·12-s − 0.554·13-s − 0.267·14-s − 1/4·16-s − 0.970·17-s + 0.471·18-s − 0.917·19-s − 0.218·21-s + 1.25·23-s − 0.612·24-s + 0.392·26-s + 0.962·27-s − 0.188·28-s + 1.67·29-s − 0.883·32-s + 0.685·34-s + 1/3·36-s − 0.493·37-s + 0.648·38-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−16.06470829087515, −15.13673473969738, −14.99565957043846, −14.11088327871114, −13.77153404225620, −13.11658016925404, −12.56872393010433, −11.92708005815626, −11.32770070025476, −10.88016417063063, −10.28579263888440, −9.890302141848100, −8.961412760915759, −8.624163956456061, −8.289644234098042, −7.444619537808172, −6.708972994641161, −6.374566778389376, −5.242492842975277, −4.951135495017660, −4.445083882241994, −3.476517252796566, −2.598959237444379, −1.768159086868842, −0.7844046513724945, 0,
0.7844046513724945, 1.768159086868842, 2.598959237444379, 3.476517252796566, 4.445083882241994, 4.951135495017660, 5.242492842975277, 6.374566778389376, 6.708972994641161, 7.444619537808172, 8.289644234098042, 8.624163956456061, 8.961412760915759, 9.890302141848100, 10.28579263888440, 10.88016417063063, 11.32770070025476, 11.92708005815626, 12.56872393010433, 13.11658016925404, 13.77153404225620, 14.11088327871114, 14.99565957043846, 15.13673473969738, 16.06470829087515
