L(s) = 1 | − 7-s + 4·11-s + 2·13-s − 6·17-s − 4·19-s + 2·29-s − 6·37-s − 2·41-s − 4·43-s + 49-s + 6·53-s + 12·59-s − 2·61-s + 4·67-s + 6·73-s − 4·77-s + 16·79-s + 12·83-s + 14·89-s − 2·91-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 6·119-s + ⋯ |
L(s) = 1 | − 0.377·7-s + 1.20·11-s + 0.554·13-s − 1.45·17-s − 0.917·19-s + 0.371·29-s − 0.986·37-s − 0.312·41-s − 0.609·43-s + 1/7·49-s + 0.824·53-s + 1.56·59-s − 0.256·61-s + 0.488·67-s + 0.702·73-s − 0.455·77-s + 1.80·79-s + 1.31·83-s + 1.48·89-s − 0.209·91-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.550·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−15.58144382988073, −15.07442114127353, −14.66309593439947, −13.94205766320361, −13.38891072143015, −13.16332130780271, −12.13676605420609, −12.09693572791854, −11.18630442761338, −10.83747812945218, −10.22899643999726, −9.494865523244120, −9.002205001855197, −8.563603797439101, −8.016950795382935, −6.981243155296097, −6.599779604825782, −6.331336247317056, −5.383394448112266, −4.732976054646407, −3.831863049849692, −3.756732557219429, −2.575776140465958, −1.957435982243902, −1.061308266347777, 0,
1.061308266347777, 1.957435982243902, 2.575776140465958, 3.756732557219429, 3.831863049849692, 4.732976054646407, 5.383394448112266, 6.331336247317056, 6.599779604825782, 6.981243155296097, 8.016950795382935, 8.563603797439101, 9.002205001855197, 9.494865523244120, 10.22899643999726, 10.83747812945218, 11.18630442761338, 12.09693572791854, 12.13676605420609, 13.16332130780271, 13.38891072143015, 13.94205766320361, 14.66309593439947, 15.07442114127353, 15.58144382988073