| L(s) = 1 | − 3-s + 3·5-s − 5·7-s − 3·11-s + 8·13-s − 3·15-s + 4·19-s + 5·21-s + 5·25-s + 27-s − 18·29-s − 31-s + 3·33-s − 15·35-s + 8·37-s − 8·39-s − 20·43-s − 6·47-s + 18·49-s − 3·53-s − 9·55-s − 4·57-s − 3·59-s − 10·61-s + 24·65-s + 10·67-s + 12·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s − 1.88·7-s − 0.904·11-s + 2.21·13-s − 0.774·15-s + 0.917·19-s + 1.09·21-s + 25-s + 0.192·27-s − 3.34·29-s − 0.179·31-s + 0.522·33-s − 2.53·35-s + 1.31·37-s − 1.28·39-s − 3.04·43-s − 0.875·47-s + 18/7·49-s − 0.412·53-s − 1.21·55-s − 0.529·57-s − 0.390·59-s − 1.28·61-s + 2.97·65-s + 1.22·67-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.175915585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.175915585\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−9.847129592906555769621365340945, −9.617811925300647502603449069179, −9.191027185931119777766546501830, −8.566238297858644265930923755764, −8.460211171309675713827260941914, −7.62411236990218427561809685546, −7.37428904978834398234973024076, −6.64494406104385811027769021964, −6.48730825145589396331878131390, −6.00759532064066943862210745351, −5.80939920583384302638292252103, −5.33894426610144068278579162544, −5.10711785909175866657311552068, −4.06400188275481559516034203982, −3.71443981500927125131399719682, −3.08691029457206792552022307801, −2.97864268332861178571832407954, −1.88787079251985058206090425438, −1.55445830857693967280399678653, −0.45760126575825555831842628499,
0.45760126575825555831842628499, 1.55445830857693967280399678653, 1.88787079251985058206090425438, 2.97864268332861178571832407954, 3.08691029457206792552022307801, 3.71443981500927125131399719682, 4.06400188275481559516034203982, 5.10711785909175866657311552068, 5.33894426610144068278579162544, 5.80939920583384302638292252103, 6.00759532064066943862210745351, 6.48730825145589396331878131390, 6.64494406104385811027769021964, 7.37428904978834398234973024076, 7.62411236990218427561809685546, 8.460211171309675713827260941914, 8.566238297858644265930923755764, 9.191027185931119777766546501830, 9.617811925300647502603449069179, 9.847129592906555769621365340945
