L(s) = 1 | + 2·3-s + 4·7-s + 9-s − 6·11-s + 2·13-s − 17-s + 4·19-s + 8·21-s − 5·25-s − 4·27-s + 4·31-s − 12·33-s − 4·37-s + 4·39-s + 6·41-s − 8·43-s + 9·49-s − 2·51-s − 6·53-s + 8·57-s − 4·61-s + 4·63-s − 8·67-s + 2·73-s − 10·75-s − 24·77-s − 8·79-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1.51·7-s + 1/3·9-s − 1.80·11-s + 0.554·13-s − 0.242·17-s + 0.917·19-s + 1.74·21-s − 25-s − 0.769·27-s + 0.718·31-s − 2.08·33-s − 0.657·37-s + 0.640·39-s + 0.937·41-s − 1.21·43-s + 9/7·49-s − 0.280·51-s − 0.824·53-s + 1.05·57-s − 0.512·61-s + 0.503·63-s − 0.977·67-s + 0.234·73-s − 1.15·75-s − 2.73·77-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.864175057\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.864175057\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 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
−11.78868723809074966331144526462, −10.96199091064046403542442582980, −9.983024040967094825270124003185, −8.763699502340737285319959905301, −8.020764485917199688918114818768, −7.56004329429510605274038490355, −5.66320470137327715389850473544, −4.65035915337384466130876980903, −3.14432806107716120656228384149, −1.95520344137895444953340586126,
1.95520344137895444953340586126, 3.14432806107716120656228384149, 4.65035915337384466130876980903, 5.66320470137327715389850473544, 7.56004329429510605274038490355, 8.020764485917199688918114818768, 8.763699502340737285319959905301, 9.983024040967094825270124003185, 10.96199091064046403542442582980, 11.78868723809074966331144526462