L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s − 2·7-s − 8-s − 2·9-s − 2·10-s − 3·11-s − 12-s + 13-s + 2·14-s − 2·15-s + 16-s + 2·17-s + 2·18-s − 2·19-s + 2·20-s + 2·21-s + 3·22-s − 7·23-s + 24-s − 25-s − 26-s + 5·27-s − 2·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.632·10-s − 0.904·11-s − 0.288·12-s + 0.277·13-s + 0.534·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.471·18-s − 0.458·19-s + 0.447·20-s + 0.436·21-s + 0.639·22-s − 1.45·23-s + 0.204·24-s − 1/5·25-s − 0.196·26-s + 0.962·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 1171 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−15.61233320444516, −15.02901585773583, −14.10874430808979, −13.82161163976982, −13.37407581104061, −12.58036057482856, −12.01025891438048, −11.82170604412448, −10.89260078637107, −10.35500039269528, −10.05290311476942, −9.739136258371446, −8.761208621922999, −8.365931738100198, −7.969550143610443, −6.906746623484762, −6.500077956179483, −6.070055422233699, −5.415649922438870, −5.017458963276975, −3.907858330222773, −3.122543494332950, −2.500095161378894, −1.868288771037607, −0.7984432036065798, 0,
0.7984432036065798, 1.868288771037607, 2.500095161378894, 3.122543494332950, 3.907858330222773, 5.017458963276975, 5.415649922438870, 6.070055422233699, 6.500077956179483, 6.906746623484762, 7.969550143610443, 8.365931738100198, 8.761208621922999, 9.739136258371446, 10.05290311476942, 10.35500039269528, 10.89260078637107, 11.82170604412448, 12.01025891438048, 12.58036057482856, 13.37407581104061, 13.82161163976982, 14.10874430808979, 15.02901585773583, 15.61233320444516