L(s) = 1 | − 2·2-s − 3-s + 2·4-s − 4·5-s + 2·6-s − 4·7-s + 9-s + 8·10-s − 5·11-s − 2·12-s − 6·13-s + 8·14-s + 4·15-s − 4·16-s − 7·17-s − 2·18-s − 4·19-s − 8·20-s + 4·21-s + 10·22-s − 6·23-s + 11·25-s + 12·26-s − 27-s − 8·28-s − 29-s − 8·30-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 1.78·5-s + 0.816·6-s − 1.51·7-s + 1/3·9-s + 2.52·10-s − 1.50·11-s − 0.577·12-s − 1.66·13-s + 2.13·14-s + 1.03·15-s − 16-s − 1.69·17-s − 0.471·18-s − 0.917·19-s − 1.78·20-s + 0.872·21-s + 2.13·22-s − 1.25·23-s + 11/5·25-s + 2.35·26-s − 0.192·27-s − 1.51·28-s − 0.185·29-s − 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12279 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12279 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 4093 | \( 1 + T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 11 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 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
−17.08504909404339, −16.62527713217313, −16.07343685278927, −15.81557577461617, −15.29549728521287, −14.80667033564637, −13.52087019233196, −13.00661852828197, −12.48508311235876, −11.98597935612577, −11.29891565093244, −10.62867238987283, −10.38421871099467, −9.724916744588480, −8.982129661802103, −8.487596855237308, −7.750401889248535, −7.196512176232023, −7.011164349394375, −6.110204111065486, −5.030759469618496, −4.437991017085432, −3.716083149460321, −2.724338755782939, −2.025543159355622, 0, 0, 0,
2.025543159355622, 2.724338755782939, 3.716083149460321, 4.437991017085432, 5.030759469618496, 6.110204111065486, 7.011164349394375, 7.196512176232023, 7.750401889248535, 8.487596855237308, 8.982129661802103, 9.724916744588480, 10.38421871099467, 10.62867238987283, 11.29891565093244, 11.98597935612577, 12.48508311235876, 13.00661852828197, 13.52087019233196, 14.80667033564637, 15.29549728521287, 15.81557577461617, 16.07343685278927, 16.62527713217313, 17.08504909404339