L(s) = 1 | − 3-s − 2·5-s − 7-s + 9-s − 6·11-s − 4·13-s + 2·15-s − 17-s − 2·19-s + 21-s + 8·23-s − 25-s − 27-s − 8·31-s + 6·33-s + 2·35-s + 4·37-s + 4·39-s + 2·41-s + 12·43-s − 2·45-s − 8·47-s + 49-s + 51-s + 6·53-s + 12·55-s + 2·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.80·11-s − 1.10·13-s + 0.516·15-s − 0.242·17-s − 0.458·19-s + 0.218·21-s + 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.43·31-s + 1.04·33-s + 0.338·35-s + 0.657·37-s + 0.640·39-s + 0.312·41-s + 1.82·43-s − 0.298·45-s − 1.16·47-s + 1/7·49-s + 0.140·51-s + 0.824·53-s + 1.61·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.75428830575213, −15.21404223560807, −14.98079333885726, −14.20645725197545, −13.32972172154413, −12.95196904350624, −12.51900665681302, −12.09232851706080, −11.14540043524114, −10.98332626714037, −10.47555257459460, −9.665404668205468, −9.268497310518844, −8.389542009886203, −7.795583017664536, −7.308909239821089, −6.957693179308368, −5.962434616341057, −5.425531703165927, −4.810511234965529, −4.326655181157449, −3.407094639478581, −2.739337609765555, −2.076565794731334, −0.6763265173858771, 0,
0.6763265173858771, 2.076565794731334, 2.739337609765555, 3.407094639478581, 4.326655181157449, 4.810511234965529, 5.425531703165927, 5.962434616341057, 6.957693179308368, 7.308909239821089, 7.795583017664536, 8.389542009886203, 9.268497310518844, 9.665404668205468, 10.47555257459460, 10.98332626714037, 11.14540043524114, 12.09232851706080, 12.51900665681302, 12.95196904350624, 13.32972172154413, 14.20645725197545, 14.98079333885726, 15.21404223560807, 15.75428830575213