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
\(L(1)\) |
\(\approx\) |
\(1.613486298\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.613486298\) |
\(L(\frac{3}{2})\) |
|
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
−13.85060176874867, −13.42844121574093, −12.84008553637796, −12.27768863449600, −11.83992922135035, −11.32916233211716, −11.03293423656489, −10.34437284251684, −9.860187738832283, −9.373572405381681, −8.705535130166808, −8.415304860707914, −7.800184306231435, −7.273724081120627, −6.707736522580355, −6.218743390573136, −5.590390941659843, −5.009772511199413, −4.486535403077561, −3.886921934611114, −3.461250097607803, −2.419332030346222, −1.995699685935769, −1.421265345022610, −0.3916651850037279,
0.3916651850037279, 1.421265345022610, 1.995699685935769, 2.419332030346222, 3.461250097607803, 3.886921934611114, 4.486535403077561, 5.009772511199413, 5.590390941659843, 6.218743390573136, 6.707736522580355, 7.273724081120627, 7.800184306231435, 8.415304860707914, 8.705535130166808, 9.373572405381681, 9.860187738832283, 10.34437284251684, 11.03293423656489, 11.32916233211716, 11.83992922135035, 12.27768863449600, 12.84008553637796, 13.42844121574093, 13.85060176874867