| L(s) = 1 | + 2·2-s + 2·4-s + 4·5-s − 6·7-s + 8·10-s + 2·11-s − 12·14-s − 4·16-s − 2·17-s − 6·19-s + 8·20-s + 4·22-s + 2·23-s + 11·25-s − 12·28-s + 14·29-s − 8·32-s − 4·34-s − 24·35-s − 12·38-s + 4·44-s + 4·46-s + 14·47-s + 18·49-s + 22·50-s + 16·53-s + 8·55-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.78·5-s − 2.26·7-s + 2.52·10-s + 0.603·11-s − 3.20·14-s − 16-s − 0.485·17-s − 1.37·19-s + 1.78·20-s + 0.852·22-s + 0.417·23-s + 11/5·25-s − 2.26·28-s + 2.59·29-s − 1.41·32-s − 0.685·34-s − 4.05·35-s − 1.94·38-s + 0.603·44-s + 0.589·46-s + 2.04·47-s + 18/7·49-s + 3.11·50-s + 2.19·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.202102048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.202102048\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
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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
−10.73057355101041417394668619248, −10.41260568178774126558927420989, −9.683766580070126929823902463290, −9.394046380951027785864914227502, −9.050481160880803499857328237920, −8.769495901785871104081806800855, −8.124663599516232198582418581196, −6.99116277238255980010032444270, −6.80141375905686787510881406975, −6.44017374849058707932241328492, −6.31661779780174955120104631988, −5.64003006683396553648944949424, −5.48985654364290840828238081237, −4.53594051909741375411514032048, −4.34955945817994985974837715151, −3.54720363056628255249617564685, −3.08769866890418410376408389428, −2.45291577844336670654215150193, −2.24877343960286584302682143519, −0.841826057761572822179262196729,
0.841826057761572822179262196729, 2.24877343960286584302682143519, 2.45291577844336670654215150193, 3.08769866890418410376408389428, 3.54720363056628255249617564685, 4.34955945817994985974837715151, 4.53594051909741375411514032048, 5.48985654364290840828238081237, 5.64003006683396553648944949424, 6.31661779780174955120104631988, 6.44017374849058707932241328492, 6.80141375905686787510881406975, 6.99116277238255980010032444270, 8.124663599516232198582418581196, 8.769495901785871104081806800855, 9.050481160880803499857328237920, 9.394046380951027785864914227502, 9.683766580070126929823902463290, 10.41260568178774126558927420989, 10.73057355101041417394668619248
