L(s) = 1 | + 2·7-s + 2·11-s + 13-s − 4·17-s + 19-s + 6·23-s + 9·29-s − 4·31-s + 3·37-s + 7·41-s − 4·43-s − 9·47-s − 3·49-s − 53-s + 6·59-s − 6·61-s + 13·67-s + 11·71-s + 8·73-s + 4·77-s + 79-s − 6·89-s + 2·91-s + 10·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | + 0.755·7-s + 0.603·11-s + 0.277·13-s − 0.970·17-s + 0.229·19-s + 1.25·23-s + 1.67·29-s − 0.718·31-s + 0.493·37-s + 1.09·41-s − 0.609·43-s − 1.31·47-s − 3/7·49-s − 0.137·53-s + 0.781·59-s − 0.768·61-s + 1.58·67-s + 1.30·71-s + 0.936·73-s + 0.455·77-s + 0.112·79-s − 0.635·89-s + 0.209·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.332802967\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.332802967\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−13.89613575530880, −13.34494271619564, −12.79915711825385, −12.46625317674230, −11.67252162912634, −11.36952301484874, −10.97091834667033, −10.52312627230445, −9.711037261926236, −9.386179312816773, −8.797759112495490, −8.272921216716673, −7.966798698957404, −7.142472359461486, −6.687891532722252, −6.342193015847842, −5.536724045100780, −4.910740611472727, −4.624659974849718, −3.915449766949720, −3.299441236911225, −2.613141202704541, −1.947603643833811, −1.246383015908560, −0.6278294571943276,
0.6278294571943276, 1.246383015908560, 1.947603643833811, 2.613141202704541, 3.299441236911225, 3.915449766949720, 4.624659974849718, 4.910740611472727, 5.536724045100780, 6.342193015847842, 6.687891532722252, 7.142472359461486, 7.966798698957404, 8.272921216716673, 8.797759112495490, 9.386179312816773, 9.711037261926236, 10.52312627230445, 10.97091834667033, 11.36952301484874, 11.67252162912634, 12.46625317674230, 12.79915711825385, 13.34494271619564, 13.89613575530880