L(s) = 1 | + 5-s − 5·11-s + 5·13-s + 17-s − 5·19-s − 23-s − 4·25-s + 6·29-s − 6·31-s + 4·37-s + 7·41-s + 7·43-s − 6·47-s − 6·53-s − 5·55-s − 14·59-s + 5·65-s + 12·67-s + 4·71-s − 6·73-s + 6·79-s + 6·83-s + 85-s + 12·89-s − 5·95-s − 8·97-s + 101-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.50·11-s + 1.38·13-s + 0.242·17-s − 1.14·19-s − 0.208·23-s − 4/5·25-s + 1.11·29-s − 1.07·31-s + 0.657·37-s + 1.09·41-s + 1.06·43-s − 0.875·47-s − 0.824·53-s − 0.674·55-s − 1.82·59-s + 0.620·65-s + 1.46·67-s + 0.474·71-s − 0.702·73-s + 0.675·79-s + 0.658·83-s + 0.108·85-s + 1.27·89-s − 0.512·95-s − 0.812·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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
−13.83207031047953, −13.20093047315392, −12.96892614346725, −12.50097942583070, −11.95596301043787, −11.13397587113659, −10.82995437735252, −10.61273653429552, −9.931254847680979, −9.390227501738128, −8.991978897300274, −8.189664911967058, −8.016742202755156, −7.561543036229204, −6.612967871179944, −6.336280050714707, −5.710905682327668, −5.409232382486616, −4.597970018842096, −4.160136378477316, −3.459088429298541, −2.843922215796519, −2.244307565268603, −1.686692116713192, −0.8436861846782433, 0,
0.8436861846782433, 1.686692116713192, 2.244307565268603, 2.843922215796519, 3.459088429298541, 4.160136378477316, 4.597970018842096, 5.409232382486616, 5.710905682327668, 6.336280050714707, 6.612967871179944, 7.561543036229204, 8.016742202755156, 8.189664911967058, 8.991978897300274, 9.390227501738128, 9.931254847680979, 10.61273653429552, 10.82995437735252, 11.13397587113659, 11.95596301043787, 12.50097942583070, 12.96892614346725, 13.20093047315392, 13.83207031047953