| L(s) = 1 | − 3-s − 7-s − 2·11-s + 6·13-s + 8·17-s + 19-s + 21-s + 8·23-s + 27-s + 8·29-s − 3·31-s + 2·33-s − 37-s − 6·39-s + 12·41-s − 22·43-s + 6·47-s − 6·49-s − 8·51-s − 12·53-s − 57-s − 4·59-s + 6·61-s + 13·67-s − 8·69-s − 20·71-s − 11·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 0.603·11-s + 1.66·13-s + 1.94·17-s + 0.229·19-s + 0.218·21-s + 1.66·23-s + 0.192·27-s + 1.48·29-s − 0.538·31-s + 0.348·33-s − 0.164·37-s − 0.960·39-s + 1.87·41-s − 3.35·43-s + 0.875·47-s − 6/7·49-s − 1.12·51-s − 1.64·53-s − 0.132·57-s − 0.520·59-s + 0.768·61-s + 1.58·67-s − 0.963·69-s − 2.37·71-s − 1.28·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.033113201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.033113201\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.404867300453660491687727090464, −8.756491253430226388308794240206, −8.481586197286060665889451892708, −8.274031303280547287898977168784, −7.75881040428835068151252853549, −7.24994931249509686195318535508, −7.01362081111772859095173615222, −6.48508810262852846364315702894, −5.96596966402527894882673227731, −5.91976027743212472565729749444, −5.35265945924216109852996909503, −4.86976500263577129667346499165, −4.68342841456055938379128876223, −3.86458612992151212977773671285, −3.39885405141085957937069191497, −3.07471891287172468221521771786, −2.76403516598285403654098936602, −1.61882765419548335065548026140, −1.26499302594765782524353377093, −0.59146379977460636388211260175,
0.59146379977460636388211260175, 1.26499302594765782524353377093, 1.61882765419548335065548026140, 2.76403516598285403654098936602, 3.07471891287172468221521771786, 3.39885405141085957937069191497, 3.86458612992151212977773671285, 4.68342841456055938379128876223, 4.86976500263577129667346499165, 5.35265945924216109852996909503, 5.91976027743212472565729749444, 5.96596966402527894882673227731, 6.48508810262852846364315702894, 7.01362081111772859095173615222, 7.24994931249509686195318535508, 7.75881040428835068151252853549, 8.274031303280547287898977168784, 8.481586197286060665889451892708, 8.756491253430226388308794240206, 9.404867300453660491687727090464
