| L(s) = 1 | + 2·5-s − 5·7-s + 6·11-s − 6·13-s + 4·17-s − 5·19-s + 4·23-s + 5·25-s + 8·29-s + 7·31-s − 10·35-s + 9·37-s + 4·41-s + 2·43-s − 2·47-s + 18·49-s + 8·53-s + 12·55-s − 10·61-s − 12·65-s − 15·67-s − 12·71-s + 11·73-s − 30·77-s + 79-s + 12·83-s + 8·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.88·7-s + 1.80·11-s − 1.66·13-s + 0.970·17-s − 1.14·19-s + 0.834·23-s + 25-s + 1.48·29-s + 1.25·31-s − 1.69·35-s + 1.47·37-s + 0.624·41-s + 0.304·43-s − 0.291·47-s + 18/7·49-s + 1.09·53-s + 1.61·55-s − 1.28·61-s − 1.48·65-s − 1.83·67-s − 1.42·71-s + 1.28·73-s − 3.41·77-s + 0.112·79-s + 1.31·83-s + 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.082349622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.082349622\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 8 T + 11 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 15 T + 158 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 8 T - 25 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−9.947016432548702552357678611376, −9.769858902041848524187846930721, −9.322889715259796338036299168142, −9.247718076365400481914575016681, −8.657473803990557409709855779032, −8.205687515322217319418128667470, −7.50765665145583404753682417190, −7.05789296144251081477554203990, −6.63492407878902111299673062076, −6.51333271219226954514676867315, −5.86642245507459681783434218931, −5.77973276086698090919173261824, −4.79284101767179958778806531719, −4.49683100950254419439239739680, −3.99649392325846079452411594199, −3.24351860671157406808041880531, −2.68472768168333848800309426797, −2.57874523151774041534991501304, −1.41258495170933294239625370695, −0.70667779511792272747013891799,
0.70667779511792272747013891799, 1.41258495170933294239625370695, 2.57874523151774041534991501304, 2.68472768168333848800309426797, 3.24351860671157406808041880531, 3.99649392325846079452411594199, 4.49683100950254419439239739680, 4.79284101767179958778806531719, 5.77973276086698090919173261824, 5.86642245507459681783434218931, 6.51333271219226954514676867315, 6.63492407878902111299673062076, 7.05789296144251081477554203990, 7.50765665145583404753682417190, 8.205687515322217319418128667470, 8.657473803990557409709855779032, 9.247718076365400481914575016681, 9.322889715259796338036299168142, 9.769858902041848524187846930721, 9.947016432548702552357678611376
