| L(s) = 1 | + 57·3-s − 1.17e3·7-s + 1.06e3·9-s + 7.56e3·11-s − 5.37e3·13-s − 2.40e4·17-s + 5.12e4·19-s − 6.69e4·21-s − 5.76e4·23-s − 6.41e4·27-s + 4.70e4·29-s + 1.92e5·31-s + 4.31e5·33-s − 1.97e5·37-s − 3.06e5·39-s − 2.37e5·41-s + 6.53e5·43-s − 8.26e5·47-s + 5.54e5·49-s − 1.36e6·51-s − 5.69e5·53-s + 2.92e6·57-s − 1.50e6·59-s − 2.06e6·61-s − 1.24e6·63-s − 3.44e6·67-s − 3.28e6·69-s + ⋯ |
| L(s) = 1 | + 1.21·3-s − 1.29·7-s + 0.485·9-s + 1.71·11-s − 0.678·13-s − 1.18·17-s + 1.71·19-s − 1.57·21-s − 0.987·23-s − 0.626·27-s + 0.358·29-s + 1.15·31-s + 2.08·33-s − 0.639·37-s − 0.826·39-s − 0.538·41-s + 1.25·43-s − 1.16·47-s + 0.673·49-s − 1.44·51-s − 0.525·53-s + 2.08·57-s − 0.951·59-s − 1.16·61-s − 0.628·63-s − 1.39·67-s − 1.20·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 19 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 1174 T + p^{7} T^{2} \) |
| 11 | \( 1 - 7563 T + p^{7} T^{2} \) |
| 13 | \( 1 + 5372 T + p^{7} T^{2} \) |
| 17 | \( 1 + 1413 p T + p^{7} T^{2} \) |
| 19 | \( 1 - 51235 T + p^{7} T^{2} \) |
| 23 | \( 1 + 57618 T + p^{7} T^{2} \) |
| 29 | \( 1 - 47040 T + p^{7} T^{2} \) |
| 31 | \( 1 - 192358 T + p^{7} T^{2} \) |
| 37 | \( 1 + 197066 T + p^{7} T^{2} \) |
| 41 | \( 1 + 237723 T + p^{7} T^{2} \) |
| 43 | \( 1 - 653012 T + p^{7} T^{2} \) |
| 47 | \( 1 + 826884 T + p^{7} T^{2} \) |
| 53 | \( 1 + 569022 T + p^{7} T^{2} \) |
| 59 | \( 1 + 1501080 T + p^{7} T^{2} \) |
| 61 | \( 1 + 2068918 T + p^{7} T^{2} \) |
| 67 | \( 1 + 3444349 T + p^{7} T^{2} \) |
| 71 | \( 1 + 4121052 T + p^{7} T^{2} \) |
| 73 | \( 1 - 83653 T + p^{7} T^{2} \) |
| 79 | \( 1 + 1454030 T + p^{7} T^{2} \) |
| 83 | \( 1 - 1626567 T + p^{7} T^{2} \) |
| 89 | \( 1 - 6004335 T + p^{7} T^{2} \) |
| 97 | \( 1 + 3411746 T + p^{7} T^{2} \) |
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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
−9.346985481556482486584067294503, −9.052704081688684217724689738600, −7.84332414668275423454674602928, −6.86161453262520336080479060496, −6.09750803551509174001512348022, −4.44503658309887931829091930761, −3.45612518344335536132975703900, −2.75839247292413006825940906519, −1.49128607268099486985582980060, 0,
1.49128607268099486985582980060, 2.75839247292413006825940906519, 3.45612518344335536132975703900, 4.44503658309887931829091930761, 6.09750803551509174001512348022, 6.86161453262520336080479060496, 7.84332414668275423454674602928, 9.052704081688684217724689738600, 9.346985481556482486584067294503
