L(s) = 1 | − 4·11-s − 2·13-s + 6·17-s − 4·19-s + 8·23-s + 2·29-s + 2·37-s + 10·41-s − 4·43-s + 14·53-s + 12·59-s + 2·61-s + 4·67-s + 2·73-s − 8·79-s + 4·83-s − 6·89-s − 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 1.20·11-s − 0.554·13-s + 1.45·17-s − 0.917·19-s + 1.66·23-s + 0.371·29-s + 0.328·37-s + 1.56·41-s − 0.609·43-s + 1.92·53-s + 1.56·59-s + 0.256·61-s + 0.488·67-s + 0.234·73-s − 0.900·79-s + 0.439·83-s − 0.635·89-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.385850888\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.385850888\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
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 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.89815781090010, −13.26672171273377, −12.80095069189879, −12.65128545729610, −11.89406022722897, −11.46366701551522, −10.75828466381514, −10.51456878773741, −9.863173565895289, −9.585562850970486, −8.697842186042654, −8.438401060972699, −7.786253205245203, −7.266371295150363, −6.946825038647795, −6.098747297328303, −5.552958593460041, −5.151460466075112, −4.596591497812586, −3.903110058527918, −3.203558608903212, −2.622025729747237, −2.199031759210790, −1.118362476541536, −0.5550079227949636,
0.5550079227949636, 1.118362476541536, 2.199031759210790, 2.622025729747237, 3.203558608903212, 3.903110058527918, 4.596591497812586, 5.151460466075112, 5.552958593460041, 6.098747297328303, 6.946825038647795, 7.266371295150363, 7.786253205245203, 8.438401060972699, 8.697842186042654, 9.585562850970486, 9.863173565895289, 10.51456878773741, 10.75828466381514, 11.46366701551522, 11.89406022722897, 12.65128545729610, 12.80095069189879, 13.26672171273377, 13.89815781090010