| L(s) = 1 | + 34·3-s + 106·7-s − 1.03e3·9-s − 1.32e3·11-s + 8.82e3·13-s − 2.40e4·17-s − 4.87e3·19-s + 3.60e3·21-s + 4.66e4·23-s − 1.09e5·27-s − 1.10e5·29-s − 2.47e5·31-s − 4.50e4·33-s + 3.60e5·37-s + 3.00e5·39-s + 1.04e5·41-s − 7.13e5·43-s + 1.56e5·47-s − 8.12e5·49-s − 8.16e5·51-s − 1.06e6·53-s − 1.65e5·57-s + 8.32e5·59-s + 5.29e5·61-s − 1.09e5·63-s − 4.17e6·67-s + 1.58e6·69-s + ⋯ |
| L(s) = 1 | + 0.727·3-s + 0.116·7-s − 0.471·9-s − 0.299·11-s + 1.11·13-s − 1.18·17-s − 0.163·19-s + 0.0849·21-s + 0.799·23-s − 1.06·27-s − 0.844·29-s − 1.49·31-s − 0.218·33-s + 1.16·37-s + 0.810·39-s + 0.236·41-s − 1.36·43-s + 0.220·47-s − 0.986·49-s − 0.861·51-s − 0.983·53-s − 0.118·57-s + 0.527·59-s + 0.298·61-s − 0.0550·63-s − 1.69·67-s + 0.581·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 34 T + p^{7} T^{2} \) |
| 7 | \( 1 - 106 T + p^{7} T^{2} \) |
| 11 | \( 1 + 1324 T + p^{7} T^{2} \) |
| 13 | \( 1 - 8828 T + p^{7} T^{2} \) |
| 17 | \( 1 + 24000 T + p^{7} T^{2} \) |
| 19 | \( 1 + 4876 T + p^{7} T^{2} \) |
| 23 | \( 1 - 46646 T + p^{7} T^{2} \) |
| 29 | \( 1 + 110902 T + p^{7} T^{2} \) |
| 31 | \( 1 + 247680 T + p^{7} T^{2} \) |
| 37 | \( 1 - 360092 T + p^{7} T^{2} \) |
| 41 | \( 1 - 104402 T + p^{7} T^{2} \) |
| 43 | \( 1 + 713622 T + p^{7} T^{2} \) |
| 47 | \( 1 - 156882 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1066268 T + p^{7} T^{2} \) |
| 59 | \( 1 - 832572 T + p^{7} T^{2} \) |
| 61 | \( 1 - 529070 T + p^{7} T^{2} \) |
| 67 | \( 1 + 4174418 T + p^{7} T^{2} \) |
| 71 | \( 1 - 5176568 T + p^{7} T^{2} \) |
| 73 | \( 1 - 237976 T + p^{7} T^{2} \) |
| 79 | \( 1 + 3742736 T + p^{7} T^{2} \) |
| 83 | \( 1 + 7861886 T + p^{7} T^{2} \) |
| 89 | \( 1 - 4300854 T + p^{7} T^{2} \) |
| 97 | \( 1 + 1147792 T + p^{7} T^{2} \) |
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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
−10.90468652471678823326020103225, −9.419490761930170309008579285034, −8.713883415628370643768806774573, −7.85116942402493317781855190964, −6.59295971425392925325302747648, −5.42090826644910419518029140180, −4.00524961612221556689217980578, −2.90735454392952570004171269717, −1.69408551811248665560918015891, 0,
1.69408551811248665560918015891, 2.90735454392952570004171269717, 4.00524961612221556689217980578, 5.42090826644910419518029140180, 6.59295971425392925325302747648, 7.85116942402493317781855190964, 8.713883415628370643768806774573, 9.419490761930170309008579285034, 10.90468652471678823326020103225
