L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 2·7-s + 8-s + 9-s + 10-s − 2·11-s − 12-s + 13-s − 2·14-s − 15-s + 16-s + 18-s + 19-s + 20-s + 2·21-s − 2·22-s − 5·23-s − 24-s + 25-s + 26-s − 27-s − 2·28-s − 6·29-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 − 0.603·11-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 + 0.229·19-s + 0.223·20-s + 0.436·21-s − 0.426·22-s − 1.04·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.377·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.37556435943715, −13.14873633604503, −12.57534031301563, −12.01694781098450, −11.79948593230441, −11.21243221811293, −10.54347588486588, −10.25729320570678, −9.889793704377379, −9.296130818927069, −8.713923787083664, −7.977633973562386, −7.720096227600316, −6.752019361270902, −6.678247033696100, −6.117416895985690, −5.537930754257308, −5.190804894337733, −4.662136413824909, −3.922032537009266, −3.471077541015927, −2.930036259262723, −2.159360065435613, −1.725558052964454, −0.7921960531645321, 0,
0.7921960531645321, 1.725558052964454, 2.159360065435613, 2.930036259262723, 3.471077541015927, 3.922032537009266, 4.662136413824909, 5.190804894337733, 5.537930754257308, 6.117416895985690, 6.678247033696100, 6.752019361270902, 7.720096227600316, 7.977633973562386, 8.713923787083664, 9.296130818927069, 9.889793704377379, 10.25729320570678, 10.54347588486588, 11.21243221811293, 11.79948593230441, 12.01694781098450, 12.57534031301563, 13.14873633604503, 13.37556435943715