| L(s) = 1 | − 2·3-s − 4-s − 7-s + 9-s + 2·12-s + 2·13-s + 16-s + 2·21-s + 2·25-s + 4·27-s + 28-s + 8·31-s − 36-s + 16·37-s − 4·39-s − 14·43-s − 2·48-s + 49-s − 2·52-s − 12·61-s − 63-s − 64-s + 73-s − 4·75-s − 4·79-s − 11·81-s − 2·84-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s − 0.377·7-s + 1/3·9-s + 0.577·12-s + 0.554·13-s + 1/4·16-s + 0.436·21-s + 2/5·25-s + 0.769·27-s + 0.188·28-s + 1.43·31-s − 1/6·36-s + 2.63·37-s − 0.640·39-s − 2.13·43-s − 0.288·48-s + 1/7·49-s − 0.277·52-s − 1.53·61-s − 0.125·63-s − 1/8·64-s + 0.117·73-s − 0.461·75-s − 0.450·79-s − 1.22·81-s − 0.218·84-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9379004215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9379004215\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.206327212375511730844697930987, −7.77182889185260405621520378174, −7.22417146473227191200512150183, −6.68091203190693064614394892984, −6.23460302623036220826637152780, −6.05478025149282823923822026226, −5.58192981977071434419273498702, −4.90783334524636256510595043651, −4.59016733721661353006644054030, −4.26314361653459966825747525633, −3.31275994520441233914210212452, −3.09685308937199660948416070339, −2.22078472910662239456073336620, −1.21929498383852468040574999783, −0.56350505744805430499663274302,
0.56350505744805430499663274302, 1.21929498383852468040574999783, 2.22078472910662239456073336620, 3.09685308937199660948416070339, 3.31275994520441233914210212452, 4.26314361653459966825747525633, 4.59016733721661353006644054030, 4.90783334524636256510595043651, 5.58192981977071434419273498702, 6.05478025149282823923822026226, 6.23460302623036220826637152780, 6.68091203190693064614394892984, 7.22417146473227191200512150183, 7.77182889185260405621520378174, 8.206327212375511730844697930987
