| L(s) = 1 | + 5-s + 7-s + 5·13-s − 6·17-s − 5·19-s − 3·23-s + 25-s − 8·31-s + 35-s + 2·37-s + 3·41-s + 4·43-s − 9·47-s − 6·49-s − 9·53-s − 15·59-s − 4·61-s + 5·65-s + 4·67-s − 6·71-s + 14·73-s − 14·79-s + 6·83-s − 6·85-s + 18·89-s + 5·91-s − 5·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 1.38·13-s − 1.45·17-s − 1.14·19-s − 0.625·23-s + 1/5·25-s − 1.43·31-s + 0.169·35-s + 0.328·37-s + 0.468·41-s + 0.609·43-s − 1.31·47-s − 6/7·49-s − 1.23·53-s − 1.95·59-s − 0.512·61-s + 0.620·65-s + 0.488·67-s − 0.712·71-s + 1.63·73-s − 1.57·79-s + 0.658·83-s − 0.650·85-s + 1.90·89-s + 0.524·91-s − 0.512·95-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 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
| 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
−7.84432176027030287302822162519, −6.72456519138276851336985368325, −6.31835275151028214959559083929, −5.66412956945919221887372184141, −4.67013580671224960260521590408, −4.12204527981708329211245796634, −3.20778625721106587787302194788, −2.10732993423136178554780902677, −1.52187493625375425037079741577, 0,
1.52187493625375425037079741577, 2.10732993423136178554780902677, 3.20778625721106587787302194788, 4.12204527981708329211245796634, 4.67013580671224960260521590408, 5.66412956945919221887372184141, 6.31835275151028214959559083929, 6.72456519138276851336985368325, 7.84432176027030287302822162519
