| L(s) = 1 | − 6·2-s − 42·3-s − 92·4-s − 84·5-s + 252·6-s + 343·7-s + 1.32e3·8-s − 423·9-s + 504·10-s − 5.56e3·11-s + 3.86e3·12-s − 5.15e3·13-s − 2.05e3·14-s + 3.52e3·15-s + 3.85e3·16-s − 1.39e4·17-s + 2.53e3·18-s + 5.53e4·19-s + 7.72e3·20-s − 1.44e4·21-s + 3.34e4·22-s − 9.12e4·23-s − 5.54e4·24-s − 7.10e4·25-s + 3.09e4·26-s + 1.09e5·27-s − 3.15e4·28-s + ⋯ |
| L(s) = 1 | − 0.530·2-s − 0.898·3-s − 0.718·4-s − 0.300·5-s + 0.476·6-s + 0.377·7-s + 0.911·8-s − 0.193·9-s + 0.159·10-s − 1.26·11-s + 0.645·12-s − 0.650·13-s − 0.200·14-s + 0.269·15-s + 0.235·16-s − 0.690·17-s + 0.102·18-s + 1.85·19-s + 0.216·20-s − 0.339·21-s + 0.668·22-s − 1.56·23-s − 0.818·24-s − 0.909·25-s + 0.344·26-s + 1.07·27-s − 0.271·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - p^{3} T \) |
| good | 2 | \( 1 + 3 p T + p^{7} T^{2} \) |
| 3 | \( 1 + 14 p T + p^{7} T^{2} \) |
| 5 | \( 1 + 84 T + p^{7} T^{2} \) |
| 11 | \( 1 + 5568 T + p^{7} T^{2} \) |
| 13 | \( 1 + 5152 T + p^{7} T^{2} \) |
| 17 | \( 1 + 13986 T + p^{7} T^{2} \) |
| 19 | \( 1 - 55370 T + p^{7} T^{2} \) |
| 23 | \( 1 + 91272 T + p^{7} T^{2} \) |
| 29 | \( 1 - 41610 T + p^{7} T^{2} \) |
| 31 | \( 1 - 150332 T + p^{7} T^{2} \) |
| 37 | \( 1 + 136366 T + p^{7} T^{2} \) |
| 41 | \( 1 + 510258 T + p^{7} T^{2} \) |
| 43 | \( 1 + 172072 T + p^{7} T^{2} \) |
| 47 | \( 1 + 519036 T + p^{7} T^{2} \) |
| 53 | \( 1 + 59202 T + p^{7} T^{2} \) |
| 59 | \( 1 - 1979250 T + p^{7} T^{2} \) |
| 61 | \( 1 + 2988748 T + p^{7} T^{2} \) |
| 67 | \( 1 - 2409404 T + p^{7} T^{2} \) |
| 71 | \( 1 - 1504512 T + p^{7} T^{2} \) |
| 73 | \( 1 + 1821022 T + p^{7} T^{2} \) |
| 79 | \( 1 + 1669240 T + p^{7} T^{2} \) |
| 83 | \( 1 - 696738 T + p^{7} T^{2} \) |
| 89 | \( 1 - 5558490 T + p^{7} T^{2} \) |
| 97 | \( 1 - 101822 p T + p^{7} T^{2} \) |
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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
−19.99031417231193878713582317681, −18.27422586042344500560304503677, −17.51282086622537592125300609925, −15.96651848563180379064634563474, −13.77648746942372162913815219723, −11.81628169961814651769762678108, −10.11800761391463716378099581502, −7.978126697923222804343204223984, −5.12939766677151346415201112010, 0,
5.12939766677151346415201112010, 7.978126697923222804343204223984, 10.11800761391463716378099581502, 11.81628169961814651769762678108, 13.77648746942372162913815219723, 15.96651848563180379064634563474, 17.51282086622537592125300609925, 18.27422586042344500560304503677, 19.99031417231193878713582317681
