| L(s) = 1 | − 4·2-s + 16·4-s − 66·5-s − 64·8-s + 264·10-s + 60·11-s + 658·13-s + 256·16-s − 414·17-s − 956·19-s − 1.05e3·20-s − 240·22-s − 600·23-s + 1.23e3·25-s − 2.63e3·26-s − 5.57e3·29-s + 3.59e3·31-s − 1.02e3·32-s + 1.65e3·34-s − 8.45e3·37-s + 3.82e3·38-s + 4.22e3·40-s + 1.91e4·41-s + 1.33e4·43-s + 960·44-s + 2.40e3·46-s − 1.96e4·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.18·5-s − 0.353·8-s + 0.834·10-s + 0.149·11-s + 1.07·13-s + 1/4·16-s − 0.347·17-s − 0.607·19-s − 0.590·20-s − 0.105·22-s − 0.236·23-s + 0.393·25-s − 0.763·26-s − 1.23·29-s + 0.671·31-s − 0.176·32-s + 0.245·34-s − 1.01·37-s + 0.429·38-s + 0.417·40-s + 1.78·41-s + 1.09·43-s + 0.0747·44-s + 0.167·46-s − 1.29·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 66 T + p^{5} T^{2} \) |
| 11 | \( 1 - 60 T + p^{5} T^{2} \) |
| 13 | \( 1 - 658 T + p^{5} T^{2} \) |
| 17 | \( 1 + 414 T + p^{5} T^{2} \) |
| 19 | \( 1 + 956 T + p^{5} T^{2} \) |
| 23 | \( 1 + 600 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5574 T + p^{5} T^{2} \) |
| 31 | \( 1 - 3592 T + p^{5} T^{2} \) |
| 37 | \( 1 + 8458 T + p^{5} T^{2} \) |
| 41 | \( 1 - 19194 T + p^{5} T^{2} \) |
| 43 | \( 1 - 13316 T + p^{5} T^{2} \) |
| 47 | \( 1 + 19680 T + p^{5} T^{2} \) |
| 53 | \( 1 - 31266 T + p^{5} T^{2} \) |
| 59 | \( 1 - 26340 T + p^{5} T^{2} \) |
| 61 | \( 1 - 31090 T + p^{5} T^{2} \) |
| 67 | \( 1 + 16804 T + p^{5} T^{2} \) |
| 71 | \( 1 + 6120 T + p^{5} T^{2} \) |
| 73 | \( 1 - 25558 T + p^{5} T^{2} \) |
| 79 | \( 1 - 74408 T + p^{5} T^{2} \) |
| 83 | \( 1 + 6468 T + p^{5} T^{2} \) |
| 89 | \( 1 + 32742 T + p^{5} T^{2} \) |
| 97 | \( 1 + 166082 T + p^{5} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.788317713073242145126141044274, −8.232003030954570257398363977036, −7.44070269273529435738327139510, −6.60552478226146103796121319796, −5.63819516786997388670113332376, −4.21600286925018025853993283934, −3.60060344928594862607804841977, −2.27451605519069113703409672629, −0.999716155244759428928960623007, 0,
0.999716155244759428928960623007, 2.27451605519069113703409672629, 3.60060344928594862607804841977, 4.21600286925018025853993283934, 5.63819516786997388670113332376, 6.60552478226146103796121319796, 7.44070269273529435738327139510, 8.232003030954570257398363977036, 8.788317713073242145126141044274
