L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s + 2·7-s + 8-s + 9-s − 2·10-s + 11-s − 12-s + 2·14-s + 2·15-s + 16-s − 17-s + 18-s + 2·19-s − 2·20-s − 2·21-s + 22-s − 6·23-s − 24-s − 25-s − 27-s + 2·28-s + 2·29-s + 2·30-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 + 1/3·9-s − 0.632·10-s + 0.301·11-s − 0.288·12-s + 0.534·14-s + 0.516·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.458·19-s − 0.447·20-s − 0.436·21-s + 0.213·22-s − 1.25·23-s − 0.204·24-s − 1/5·25-s − 0.192·27-s + 0.377·28-s + 0.371·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.36617398236076, −12.78828160085700, −12.24477130207090, −11.87319077946192, −11.62306866002762, −11.24460383099493, −10.64722325683169, −10.24774482803839, −9.667559581231790, −9.099559322590109, −8.299349865521237, −8.027548597717235, −7.658358758238872, −6.900604985153844, −6.641364867049570, −5.984253214657607, −5.478999822069936, −4.926162667947444, −4.495492324621473, −4.064048005378783, −3.457838585635134, −3.018220278252665, −1.929475407480767, −1.741005864159193, −0.7794103014452556, 0,
0.7794103014452556, 1.741005864159193, 1.929475407480767, 3.018220278252665, 3.457838585635134, 4.064048005378783, 4.495492324621473, 4.926162667947444, 5.478999822069936, 5.984253214657607, 6.641364867049570, 6.900604985153844, 7.658358758238872, 8.027548597717235, 8.299349865521237, 9.099559322590109, 9.667559581231790, 10.24774482803839, 10.64722325683169, 11.24460383099493, 11.62306866002762, 11.87319077946192, 12.24477130207090, 12.78828160085700, 13.36617398236076