| L(s) = 1 | + 3-s + 9-s − 4·11-s + 2·13-s + 6·17-s − 4·19-s − 23-s + 27-s − 2·29-s − 4·33-s + 2·37-s + 2·39-s + 10·41-s − 4·43-s − 7·49-s + 6·51-s − 6·53-s − 4·57-s + 4·59-s − 10·61-s − 12·67-s − 69-s + 8·71-s − 10·73-s + 8·79-s + 81-s − 4·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s − 0.208·23-s + 0.192·27-s − 0.371·29-s − 0.696·33-s + 0.328·37-s + 0.320·39-s + 1.56·41-s − 0.609·43-s − 49-s + 0.840·51-s − 0.824·53-s − 0.529·57-s + 0.520·59-s − 1.28·61-s − 1.46·67-s − 0.120·69-s + 0.949·71-s − 1.17·73-s + 0.900·79-s + 1/9·81-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−15.46442540487089, −14.90836890850573, −14.48158874142206, −13.98931716815860, −13.18764364880755, −13.06909278948686, −12.39583249053382, −11.85480053449703, −11.05472554183195, −10.60253520422074, −10.14096169864558, −9.494682306984134, −8.978664304867901, −8.247582290301471, −7.788348618630399, −7.528673283064173, −6.523602574608811, −6.009270273079069, −5.359557178934098, −4.697020093415687, −3.987090360204787, −3.276674191128235, −2.745126239943759, −1.954020232227422, −1.142635890852120, 0,
1.142635890852120, 1.954020232227422, 2.745126239943759, 3.276674191128235, 3.987090360204787, 4.697020093415687, 5.359557178934098, 6.009270273079069, 6.523602574608811, 7.528673283064173, 7.788348618630399, 8.247582290301471, 8.978664304867901, 9.494682306984134, 10.14096169864558, 10.60253520422074, 11.05472554183195, 11.85480053449703, 12.39583249053382, 13.06909278948686, 13.18764364880755, 13.98931716815860, 14.48158874142206, 14.90836890850573, 15.46442540487089
