| L(s) = 1 | + 3·7-s + 3·11-s + 13-s − 7·17-s − 8·19-s − 4·23-s + 3·29-s − 11·31-s + 2·41-s − 8·43-s + 9·47-s + 2·49-s − 9·53-s + 9·59-s + 61-s + 5·67-s − 12·73-s + 9·77-s − 8·79-s + 9·83-s − 12·89-s + 3·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.13·7-s + 0.904·11-s + 0.277·13-s − 1.69·17-s − 1.83·19-s − 0.834·23-s + 0.557·29-s − 1.97·31-s + 0.312·41-s − 1.21·43-s + 1.31·47-s + 2/7·49-s − 1.23·53-s + 1.17·59-s + 0.128·61-s + 0.610·67-s − 1.40·73-s + 1.02·77-s − 0.900·79-s + 0.987·83-s − 1.27·89-s + 0.314·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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
| \(L(1)\) |
\(\approx\) |
\(1.665958989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.665958989\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 11 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−13.92794369641442, −13.23579015209575, −12.93564571519671, −12.36807817629827, −11.71021323809928, −11.37274872911013, −10.88270712996729, −10.57543117033944, −9.898584736915974, −9.070490410094848, −8.896356380530837, −8.342135399485654, −7.964813742247407, −7.093792122673808, −6.795012037636590, −6.172004615587588, −5.705967722566154, −4.908832813594992, −4.358106704340928, −4.110825124190936, −3.422203067942595, −2.366831354307683, −1.950512717386383, −1.500431084005232, −0.3906234923297209,
0.3906234923297209, 1.500431084005232, 1.950512717386383, 2.366831354307683, 3.422203067942595, 4.110825124190936, 4.358106704340928, 4.908832813594992, 5.705967722566154, 6.172004615587588, 6.795012037636590, 7.093792122673808, 7.964813742247407, 8.342135399485654, 8.896356380530837, 9.070490410094848, 9.898584736915974, 10.57543117033944, 10.88270712996729, 11.37274872911013, 11.71021323809928, 12.36807817629827, 12.93564571519671, 13.23579015209575, 13.92794369641442
