| L(s) = 1 | − 7-s + 11-s − 6·13-s − 5·17-s + 2·19-s + 2·23-s + 6·29-s + 3·31-s + 5·37-s + 6·41-s − 11·43-s + 47-s + 49-s + 53-s − 15·59-s + 10·61-s − 10·67-s − 10·71-s − 5·73-s − 77-s − 17·79-s + 12·83-s + 2·89-s + 6·91-s + 12·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.301·11-s − 1.66·13-s − 1.21·17-s + 0.458·19-s + 0.417·23-s + 1.11·29-s + 0.538·31-s + 0.821·37-s + 0.937·41-s − 1.67·43-s + 0.145·47-s + 1/7·49-s + 0.137·53-s − 1.95·59-s + 1.28·61-s − 1.22·67-s − 1.18·71-s − 0.585·73-s − 0.113·77-s − 1.91·79-s + 1.31·83-s + 0.211·89-s + 0.628·91-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.236557159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.236557159\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−14.25208628420347, −13.52662976209505, −13.22524336546358, −12.64593552366232, −12.09040178811806, −11.73228475425663, −11.23577673765717, −10.50130630043792, −10.06514850478148, −9.657204608272074, −9.013561900558611, −8.699283425455398, −7.891304501263939, −7.385450236978627, −6.945352268546833, −6.341971364324956, −5.908490584045490, −4.952965249789787, −4.701883643244039, −4.157210507565362, −3.198501635058853, −2.756426960120640, −2.175267553154619, −1.295763768953486, −0.3762012130123717,
0.3762012130123717, 1.295763768953486, 2.175267553154619, 2.756426960120640, 3.198501635058853, 4.157210507565362, 4.701883643244039, 4.952965249789787, 5.908490584045490, 6.341971364324956, 6.945352268546833, 7.385450236978627, 7.891304501263939, 8.699283425455398, 9.013561900558611, 9.657204608272074, 10.06514850478148, 10.50130630043792, 11.23577673765717, 11.73228475425663, 12.09040178811806, 12.64593552366232, 13.22524336546358, 13.52662976209505, 14.25208628420347
