| L(s) = 1 | − 5-s − 11-s + 7·19-s − 6·23-s + 25-s − 7·29-s − 31-s − 2·37-s + 9·41-s + 6·43-s + 2·47-s − 7·49-s + 55-s − 3·59-s − 10·61-s + 2·67-s − 71-s − 4·79-s + 6·83-s − 7·89-s − 7·95-s + 2·97-s − 9·101-s + 6·103-s + 2·107-s + 3·109-s + 6·115-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s + 1.60·19-s − 1.25·23-s + 1/5·25-s − 1.29·29-s − 0.179·31-s − 0.328·37-s + 1.40·41-s + 0.914·43-s + 0.291·47-s − 49-s + 0.134·55-s − 0.390·59-s − 1.28·61-s + 0.244·67-s − 0.118·71-s − 0.450·79-s + 0.658·83-s − 0.741·89-s − 0.718·95-s + 0.203·97-s − 0.895·101-s + 0.591·103-s + 0.193·107-s + 0.287·109-s + 0.559·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 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
−7.61816352204952883606547561010, −7.21557864639319702819152027082, −6.11890657007085211672885982622, −5.60178467867895649486925002917, −4.77361653867230127300700190913, −3.94815618069838174830052295662, −3.27408207619664390123385305395, −2.34424077102232550887640075609, −1.26325184500356442181158965717, 0,
1.26325184500356442181158965717, 2.34424077102232550887640075609, 3.27408207619664390123385305395, 3.94815618069838174830052295662, 4.77361653867230127300700190913, 5.60178467867895649486925002917, 6.11890657007085211672885982622, 7.21557864639319702819152027082, 7.61816352204952883606547561010
