| L(s) = 1 | + 2-s − 3-s + 4-s − 4·5-s − 6-s − 2·7-s + 8-s + 9-s − 4·10-s − 11-s − 12-s − 4·13-s − 2·14-s + 4·15-s + 16-s + 2·17-s + 18-s − 4·20-s + 2·21-s − 22-s + 6·23-s − 24-s + 11·25-s − 4·26-s − 27-s − 2·28-s − 10·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 1.26·10-s − 0.301·11-s − 0.288·12-s − 1.10·13-s − 0.534·14-s + 1.03·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.894·20-s + 0.436·21-s − 0.213·22-s + 1.25·23-s − 0.204·24-s + 11/5·25-s − 0.784·26-s − 0.192·27-s − 0.377·28-s − 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63426 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63426 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.79309689022259, −14.08377408370708, −13.15061882107657, −12.90596754828971, −12.53158695291912, −11.99858276233161, −11.55413934676464, −11.11657602867110, −10.75401840706882, −9.933182463002459, −9.564019517651671, −8.795567588307715, −8.100719223045381, −7.558611443741876, −7.155189777796255, −6.852233638480389, −6.020583952832621, −5.398278860759427, −4.877283789639009, −4.376224798062634, −3.764929218002493, −3.198081905756297, −2.793476119397722, −1.722192266911329, −0.6576137725690980, 0,
0.6576137725690980, 1.722192266911329, 2.793476119397722, 3.198081905756297, 3.764929218002493, 4.376224798062634, 4.877283789639009, 5.398278860759427, 6.020583952832621, 6.852233638480389, 7.155189777796255, 7.558611443741876, 8.100719223045381, 8.795567588307715, 9.564019517651671, 9.933182463002459, 10.75401840706882, 11.11657602867110, 11.55413934676464, 11.99858276233161, 12.53158695291912, 12.90596754828971, 13.15061882107657, 14.08377408370708, 14.79309689022259
