L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 2·7-s − 8-s + 9-s − 10-s − 12-s − 13-s + 2·14-s − 15-s + 16-s − 18-s + 5·19-s + 20-s + 2·21-s + 4·23-s + 24-s + 25-s + 26-s − 27-s − 2·28-s + 29-s + 30-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.277·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s + 1.14·19-s + 0.223·20-s + 0.436·21-s + 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.377·28-s + 0.185·29-s + 0.182·30-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.02767976643761, −14.29692782559090, −13.79295445136976, −13.24099042979992, −12.68138402959967, −12.23323747547052, −11.73655918748498, −11.06911936412688, −10.71439270828742, −10.06305097108275, −9.639187830678804, −9.234764163604087, −8.736127739794655, −7.818372572396700, −7.469631652665544, −6.837097607899543, −6.286305632713912, −5.894926385003506, −5.117086318292309, −4.699995882710703, −3.665239713425295, −3.077952403466306, −2.468547759044970, −1.534242053305360, −0.8967781355555587, 0,
0.8967781355555587, 1.534242053305360, 2.468547759044970, 3.077952403466306, 3.665239713425295, 4.699995882710703, 5.117086318292309, 5.894926385003506, 6.286305632713912, 6.837097607899543, 7.469631652665544, 7.818372572396700, 8.736127739794655, 9.234764163604087, 9.639187830678804, 10.06305097108275, 10.71439270828742, 11.06911936412688, 11.73655918748498, 12.23323747547052, 12.68138402959967, 13.24099042979992, 13.79295445136976, 14.29692782559090, 15.02767976643761