| L(s) = 1 | + 3-s − 2·4-s − 5-s − 3·7-s + 9-s − 4·11-s − 2·12-s − 15-s + 4·16-s − 3·17-s + 5·19-s + 2·20-s − 3·21-s + 23-s + 25-s + 27-s + 6·28-s − 29-s + 8·31-s − 4·33-s + 3·35-s − 2·36-s + 4·37-s − 6·41-s + 4·43-s + 8·44-s − 45-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s − 1.13·7-s + 1/3·9-s − 1.20·11-s − 0.577·12-s − 0.258·15-s + 16-s − 0.727·17-s + 1.14·19-s + 0.447·20-s − 0.654·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.13·28-s − 0.185·29-s + 1.43·31-s − 0.696·33-s + 0.507·35-s − 1/3·36-s + 0.657·37-s − 0.937·41-s + 0.609·43-s + 1.20·44-s − 0.149·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−16.83933556450473, −16.25874902902652, −15.62869473666497, −15.39377605226625, −14.60105478549672, −13.88201240914562, −13.41433523795680, −13.08065237832338, −12.50156213526060, −11.85500715385217, −10.98889857675423, −10.27393690024541, −9.688212498124283, −9.409609161753552, −8.466023174974830, −8.182135288181352, −7.409463722517379, −6.767742999994972, −5.892066868807489, −5.123844612937545, −4.519762130975307, −3.696064517701593, −3.107351547206115, −2.456465682567805, −0.9582095215968572, 0,
0.9582095215968572, 2.456465682567805, 3.107351547206115, 3.696064517701593, 4.519762130975307, 5.123844612937545, 5.892066868807489, 6.767742999994972, 7.409463722517379, 8.182135288181352, 8.466023174974830, 9.409609161753552, 9.688212498124283, 10.27393690024541, 10.98889857675423, 11.85500715385217, 12.50156213526060, 13.08065237832338, 13.41433523795680, 13.88201240914562, 14.60105478549672, 15.39377605226625, 15.62869473666497, 16.25874902902652, 16.83933556450473
