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
\(L(4)\) |
\(\approx\) |
\(1.672544965\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.672544965\) |
\(L(\frac{9}{2})\) |
|
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
−10.99843050144852370980181120826, −9.852957815454752252805753852197, −9.087067316046095081282621540862, −7.970656211554168462769432015843, −7.29201720048977324040212141281, −5.32365828149134613629774041817, −4.96218272446456237936666739288, −3.66562642480412426145853828502, −2.03243167146514061235406675266, −0.68744509686538829116542154625,
0.68744509686538829116542154625, 2.03243167146514061235406675266, 3.66562642480412426145853828502, 4.96218272446456237936666739288, 5.32365828149134613629774041817, 7.29201720048977324040212141281, 7.970656211554168462769432015843, 9.087067316046095081282621540862, 9.852957815454752252805753852197, 10.99843050144852370980181120826