L(s) = 1 | − 128·4-s − 1.25e3·7-s + 1.26e4·13-s + 1.63e4·16-s + 4.30e4·19-s + 1.60e5·28-s − 3.31e5·31-s + 2.79e5·37-s − 4.09e5·43-s + 7.51e5·49-s − 1.61e6·52-s + 1.99e6·61-s − 2.09e6·64-s − 4.05e6·67-s − 6.27e6·73-s − 5.51e6·76-s + 8.76e6·79-s − 1.58e7·91-s − 1.75e7·97-s + 8.02e6·103-s + 2.67e7·109-s − 2.05e7·112-s + ⋯ |
L(s) = 1 | − 4-s − 1.38·7-s + 1.59·13-s + 16-s + 1.44·19-s + 1.38·28-s − 1.99·31-s + 0.907·37-s − 0.785·43-s + 0.912·49-s − 1.59·52-s + 1.12·61-s − 64-s − 1.64·67-s − 1.88·73-s − 1.44·76-s + 1.99·79-s − 2.20·91-s − 1.94·97-s + 0.723·103-s + 1.97·109-s − 1.38·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + p^{7} T^{2} \) |
| 7 | \( 1 + 1255 T + p^{7} T^{2} \) |
| 11 | \( 1 + p^{7} T^{2} \) |
| 13 | \( 1 - 12605 T + p^{7} T^{2} \) |
| 17 | \( 1 + p^{7} T^{2} \) |
| 19 | \( 1 - 43091 T + p^{7} T^{2} \) |
| 23 | \( 1 + p^{7} T^{2} \) |
| 29 | \( 1 + p^{7} T^{2} \) |
| 31 | \( 1 + 331387 T + p^{7} T^{2} \) |
| 37 | \( 1 - 279710 T + p^{7} T^{2} \) |
| 41 | \( 1 + p^{7} T^{2} \) |
| 43 | \( 1 + 409495 T + p^{7} T^{2} \) |
| 47 | \( 1 + p^{7} T^{2} \) |
| 53 | \( 1 + p^{7} T^{2} \) |
| 59 | \( 1 + p^{7} T^{2} \) |
| 61 | \( 1 - 1998347 T + p^{7} T^{2} \) |
| 67 | \( 1 + 4058455 T + p^{7} T^{2} \) |
| 71 | \( 1 + p^{7} T^{2} \) |
| 73 | \( 1 + 6274810 T + p^{7} T^{2} \) |
| 79 | \( 1 - 8763044 T + p^{7} T^{2} \) |
| 83 | \( 1 + p^{7} T^{2} \) |
| 89 | \( 1 + p^{7} T^{2} \) |
| 97 | \( 1 + 17521555 T + p^{7} T^{2} \) |
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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
−10.28880800272055627250871987482, −9.419966699225755045474149409977, −8.788407478919382364114776182057, −7.54107496843527487671215138755, −6.25980507486084562642765308659, −5.38750611255653214437617153415, −3.87893278591502287082528764272, −3.21304665889927488752247904259, −1.17798257100019571982032238594, 0,
1.17798257100019571982032238594, 3.21304665889927488752247904259, 3.87893278591502287082528764272, 5.38750611255653214437617153415, 6.25980507486084562642765308659, 7.54107496843527487671215138755, 8.788407478919382364114776182057, 9.419966699225755045474149409977, 10.28880800272055627250871987482