| L(s) = 1 | − 84·3-s − 82·5-s − 456·7-s + 4.86e3·9-s − 2.52e3·11-s − 1.07e4·13-s + 6.88e3·15-s − 1.11e4·17-s + 4.12e3·19-s + 3.83e4·21-s + 8.17e4·23-s − 7.14e4·25-s − 2.25e5·27-s + 9.97e4·29-s − 4.04e4·31-s + 2.12e5·33-s + 3.73e4·35-s − 4.19e5·37-s + 9.05e5·39-s + 1.41e5·41-s − 6.90e5·43-s − 3.99e5·45-s − 6.82e5·47-s − 6.15e5·49-s + 9.36e5·51-s + 1.81e6·53-s + 2.06e5·55-s + ⋯ |
| L(s) = 1 | − 1.79·3-s − 0.293·5-s − 0.502·7-s + 2.22·9-s − 0.571·11-s − 1.36·13-s + 0.526·15-s − 0.550·17-s + 0.137·19-s + 0.902·21-s + 1.40·23-s − 0.913·25-s − 2.20·27-s + 0.759·29-s − 0.244·31-s + 1.02·33-s + 0.147·35-s − 1.36·37-s + 2.44·39-s + 0.320·41-s − 1.32·43-s − 0.653·45-s − 0.958·47-s − 0.747·49-s + 0.988·51-s + 1.67·53-s + 0.167·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 28 p T + p^{7} T^{2} \) |
| 5 | \( 1 + 82 T + p^{7} T^{2} \) |
| 7 | \( 1 + 456 T + p^{7} T^{2} \) |
| 11 | \( 1 + 2524 T + p^{7} T^{2} \) |
| 13 | \( 1 + 10778 T + p^{7} T^{2} \) |
| 17 | \( 1 + 11150 T + p^{7} T^{2} \) |
| 19 | \( 1 - 4124 T + p^{7} T^{2} \) |
| 23 | \( 1 - 81704 T + p^{7} T^{2} \) |
| 29 | \( 1 - 99798 T + p^{7} T^{2} \) |
| 31 | \( 1 + 40480 T + p^{7} T^{2} \) |
| 37 | \( 1 + 419442 T + p^{7} T^{2} \) |
| 41 | \( 1 - 141402 T + p^{7} T^{2} \) |
| 43 | \( 1 + 690428 T + p^{7} T^{2} \) |
| 47 | \( 1 + 682032 T + p^{7} T^{2} \) |
| 53 | \( 1 - 1813118 T + p^{7} T^{2} \) |
| 59 | \( 1 + 966028 T + p^{7} T^{2} \) |
| 61 | \( 1 - 1887670 T + p^{7} T^{2} \) |
| 67 | \( 1 - 2965868 T + p^{7} T^{2} \) |
| 71 | \( 1 + 2548232 T + p^{7} T^{2} \) |
| 73 | \( 1 + 1680326 T + p^{7} T^{2} \) |
| 79 | \( 1 - 4038064 T + p^{7} T^{2} \) |
| 83 | \( 1 + 5385764 T + p^{7} T^{2} \) |
| 89 | \( 1 + 6473046 T + p^{7} T^{2} \) |
| 97 | \( 1 + 6065758 T + p^{7} T^{2} \) |
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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
−19.33539488014304505369483389471, −17.84118289171870269700157743928, −16.76571096552944371160403574876, −15.51849604957907210792176341191, −12.84884691759405480078086906011, −11.61154968464071773737550398844, −10.13804380825495415909992963500, −6.93776263614106550406017956614, −5.07707747027606474477459261931, 0,
5.07707747027606474477459261931, 6.93776263614106550406017956614, 10.13804380825495415909992963500, 11.61154968464071773737550398844, 12.84884691759405480078086906011, 15.51849604957907210792176341191, 16.76571096552944371160403574876, 17.84118289171870269700157743928, 19.33539488014304505369483389471
