| L(s) = 1 | + 2·2-s − 4-s − 8·8-s − 6·11-s − 7·16-s + 4·17-s − 12·22-s + 12·29-s + 14·32-s + 8·34-s + 12·37-s + 12·41-s + 6·44-s + 12·49-s + 24·58-s + 35·64-s − 4·68-s + 24·74-s + 24·82-s + 28·83-s + 48·88-s + 24·97-s + 24·98-s + 12·101-s − 24·103-s − 28·107-s − 12·116-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s − 2.82·8-s − 1.80·11-s − 7/4·16-s + 0.970·17-s − 2.55·22-s + 2.22·29-s + 2.47·32-s + 1.37·34-s + 1.97·37-s + 1.87·41-s + 0.904·44-s + 12/7·49-s + 3.15·58-s + 35/8·64-s − 0.485·68-s + 2.78·74-s + 2.65·82-s + 3.07·83-s + 5.11·88-s + 2.43·97-s + 2.42·98-s + 1.19·101-s − 2.36·103-s − 2.70·107-s − 1.11·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.066640524\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.066640524\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 84 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 88 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
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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
−9.025006352044451319641204509237, −8.799131407351690386713777224042, −8.176784336271066436283124470925, −8.124073886751658111345164661784, −7.58368930281040510678030958622, −7.35354113104240853363871835985, −6.40450117503473910746861670789, −6.30664624240425543163561302496, −5.82817618593666922605495417032, −5.50882894715251761486367916589, −5.00131607466503119947630427428, −4.92980331900693162375297291295, −4.24255895625171122032305338978, −4.18530170196182969235863349351, −3.45706729739573989204290124727, −3.06917216995174983514905654067, −2.51181449265240366945285808389, −2.46796663130447926553305366686, −0.921289233781082248653117460526, −0.62117814207963761666542739312,
0.62117814207963761666542739312, 0.921289233781082248653117460526, 2.46796663130447926553305366686, 2.51181449265240366945285808389, 3.06917216995174983514905654067, 3.45706729739573989204290124727, 4.18530170196182969235863349351, 4.24255895625171122032305338978, 4.92980331900693162375297291295, 5.00131607466503119947630427428, 5.50882894715251761486367916589, 5.82817618593666922605495417032, 6.30664624240425543163561302496, 6.40450117503473910746861670789, 7.35354113104240853363871835985, 7.58368930281040510678030958622, 8.124073886751658111345164661784, 8.176784336271066436283124470925, 8.799131407351690386713777224042, 9.025006352044451319641204509237
