| L(s) = 1 | − 2·3-s + 7-s − 3·9-s + 6·11-s − 3·13-s − 6·17-s + 5·19-s − 2·21-s − 12·23-s + 14·27-s − 5·29-s + 6·31-s − 12·33-s + 4·37-s + 6·39-s − 6·41-s + 3·43-s + 47-s − 12·49-s + 12·51-s + 2·53-s − 10·57-s + 15·59-s + 4·61-s − 3·63-s − 14·67-s + 24·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s − 9-s + 1.80·11-s − 0.832·13-s − 1.45·17-s + 1.14·19-s − 0.436·21-s − 2.50·23-s + 2.69·27-s − 0.928·29-s + 1.07·31-s − 2.08·33-s + 0.657·37-s + 0.960·39-s − 0.937·41-s + 0.457·43-s + 0.145·47-s − 1.71·49-s + 1.68·51-s + 0.274·53-s − 1.32·57-s + 1.95·59-s + 0.512·61-s − 0.377·63-s − 1.71·67-s + 2.88·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.473600857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.473600857\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 33 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 73 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 77 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T + 93 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 87 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 15 T + 163 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 81 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 14 T + 178 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 11 T + 171 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 5 T + 133 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 177 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 183 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−7.64824961995607438276222837276, −7.58256157592927554216971391934, −7.02196972942130398463108662581, −6.61970694315756961986379481611, −6.41883430976531365165324233420, −6.15354258616454343894897143745, −5.78207333507983897116187494921, −5.61970296453562844387834001546, −4.94038330424265115831022506001, −4.92056851583725871929724767066, −4.40531544784867458299058915958, −4.11897141050590023814370146048, −3.60134250862243611148278528056, −3.37325117241713319554365943503, −2.55373460995541527467768255172, −2.54308627954451348977099643290, −1.70399031143269131512794754504, −1.64079454363098128415151813266, −0.55786222492347363933738490197, −0.51381759755279641076785231776,
0.51381759755279641076785231776, 0.55786222492347363933738490197, 1.64079454363098128415151813266, 1.70399031143269131512794754504, 2.54308627954451348977099643290, 2.55373460995541527467768255172, 3.37325117241713319554365943503, 3.60134250862243611148278528056, 4.11897141050590023814370146048, 4.40531544784867458299058915958, 4.92056851583725871929724767066, 4.94038330424265115831022506001, 5.61970296453562844387834001546, 5.78207333507983897116187494921, 6.15354258616454343894897143745, 6.41883430976531365165324233420, 6.61970694315756961986379481611, 7.02196972942130398463108662581, 7.58256157592927554216971391934, 7.64824961995607438276222837276
