| L(s) = 1 | + 240·5-s + 343·7-s − 702·11-s − 3.95e3·13-s + 3.40e3·17-s − 4.90e4·19-s + 1.15e4·23-s − 2.05e4·25-s − 4.96e4·29-s − 1.13e5·31-s + 8.23e4·35-s − 6.68e4·37-s + 3.60e5·41-s − 7.65e5·43-s + 1.34e6·47-s + 1.17e5·49-s − 3.58e5·53-s − 1.68e5·55-s − 9.30e5·59-s − 1.31e6·61-s − 9.49e5·65-s + 1.89e6·67-s − 2.27e5·71-s + 7.84e5·73-s − 2.40e5·77-s − 2.10e6·79-s − 8.62e6·83-s + ⋯ |
| L(s) = 1 | + 0.858·5-s + 0.377·7-s − 0.159·11-s − 0.499·13-s + 0.168·17-s − 1.64·19-s + 0.197·23-s − 0.262·25-s − 0.378·29-s − 0.683·31-s + 0.324·35-s − 0.217·37-s + 0.817·41-s − 1.46·43-s + 1.88·47-s + 1/7·49-s − 0.331·53-s − 0.136·55-s − 0.589·59-s − 0.743·61-s − 0.429·65-s + 0.769·67-s − 0.0755·71-s + 0.236·73-s − 0.0601·77-s − 0.479·79-s − 1.65·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p^{3} T \) |
| good | 5 | \( 1 - 48 p T + p^{7} T^{2} \) |
| 11 | \( 1 + 702 T + p^{7} T^{2} \) |
| 13 | \( 1 + 3958 T + p^{7} T^{2} \) |
| 17 | \( 1 - 3408 T + p^{7} T^{2} \) |
| 19 | \( 1 + 49036 T + p^{7} T^{2} \) |
| 23 | \( 1 - 11514 T + p^{7} T^{2} \) |
| 29 | \( 1 + 49662 T + p^{7} T^{2} \) |
| 31 | \( 1 + 113320 T + p^{7} T^{2} \) |
| 37 | \( 1 + 66886 T + p^{7} T^{2} \) |
| 41 | \( 1 - 360900 T + p^{7} T^{2} \) |
| 43 | \( 1 + 765292 T + p^{7} T^{2} \) |
| 47 | \( 1 - 1344876 T + p^{7} T^{2} \) |
| 53 | \( 1 + 358962 T + p^{7} T^{2} \) |
| 59 | \( 1 + 930528 T + p^{7} T^{2} \) |
| 61 | \( 1 + 1318834 T + p^{7} T^{2} \) |
| 67 | \( 1 - 1893464 T + p^{7} T^{2} \) |
| 71 | \( 1 + 227994 T + p^{7} T^{2} \) |
| 73 | \( 1 - 784934 T + p^{7} T^{2} \) |
| 79 | \( 1 + 2100892 T + p^{7} T^{2} \) |
| 83 | \( 1 + 8629308 T + p^{7} T^{2} \) |
| 89 | \( 1 + 5903100 T + p^{7} T^{2} \) |
| 97 | \( 1 - 773846 T + p^{7} T^{2} \) |
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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
−10.32710956583376379242033298416, −9.396063367773072340913032960223, −8.449395644226465842662660098935, −7.32387314611039918924178205954, −6.19895322353618508221340771101, −5.27237662271730489677279398532, −4.10764779194060567809777612935, −2.53696602653211729686934420842, −1.60681576002915275841966501844, 0,
1.60681576002915275841966501844, 2.53696602653211729686934420842, 4.10764779194060567809777612935, 5.27237662271730489677279398532, 6.19895322353618508221340771101, 7.32387314611039918924178205954, 8.449395644226465842662660098935, 9.396063367773072340913032960223, 10.32710956583376379242033298416
