L(s) = 1 | − 5-s − 7-s − 4·11-s − 6·13-s − 6·17-s + 4·19-s − 4·23-s + 25-s + 2·29-s − 8·31-s + 35-s + 6·37-s − 6·41-s − 8·43-s + 49-s − 6·53-s + 4·55-s − 4·59-s + 10·61-s + 6·65-s − 8·67-s + 12·71-s − 14·73-s + 4·77-s + 16·79-s − 12·83-s + 6·85-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s − 1.20·11-s − 1.66·13-s − 1.45·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.169·35-s + 0.986·37-s − 0.937·41-s − 1.21·43-s + 1/7·49-s − 0.824·53-s + 0.539·55-s − 0.520·59-s + 1.28·61-s + 0.744·65-s − 0.977·67-s + 1.42·71-s − 1.63·73-s + 0.455·77-s + 1.80·79-s − 1.31·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3895372224\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3895372224\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−16.52189860046360, −16.01494682771090, −15.53768814768427, −14.97670379680702, −14.44504817499682, −13.64428788481272, −13.15887139334858, −12.58899027177918, −12.04175371089562, −11.38858716901564, −10.83614339398544, −9.999212678763549, −9.747934068978059, −8.917011304205502, −8.194839245585873, −7.568583813101380, −7.115314795516845, −6.396407459669255, −5.442994507762571, −4.932296533399160, −4.275150445559724, −3.303046644792985, −2.612421185534779, −1.899875698996430, −0.2730217166958997,
0.2730217166958997, 1.899875698996430, 2.612421185534779, 3.303046644792985, 4.275150445559724, 4.932296533399160, 5.442994507762571, 6.396407459669255, 7.115314795516845, 7.568583813101380, 8.194839245585873, 8.917011304205502, 9.747934068978059, 9.999212678763549, 10.83614339398544, 11.38858716901564, 12.04175371089562, 12.58899027177918, 13.15887139334858, 13.64428788481272, 14.44504817499682, 14.97670379680702, 15.53768814768427, 16.01494682771090, 16.52189860046360