| L(s) = 1 | + 2·2-s + 2·4-s + 5·7-s + 9-s + 10·14-s − 4·16-s + 8·17-s + 2·18-s − 3·23-s + 4·25-s + 10·28-s − 5·31-s − 8·32-s + 16·34-s + 2·36-s + 41-s − 6·46-s − 16·47-s + 7·49-s + 8·50-s − 10·62-s + 5·63-s − 8·64-s + 16·68-s − 7·71-s + 18·73-s − 79-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.88·7-s + 1/3·9-s + 2.67·14-s − 16-s + 1.94·17-s + 0.471·18-s − 0.625·23-s + 4/5·25-s + 1.88·28-s − 0.898·31-s − 1.41·32-s + 2.74·34-s + 1/3·36-s + 0.156·41-s − 0.884·46-s − 2.33·47-s + 49-s + 1.13·50-s − 1.27·62-s + 0.629·63-s − 64-s + 1.94·68-s − 0.830·71-s + 2.10·73-s − 0.112·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.387288519\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.387288519\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 2473 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 71 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 36 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 104 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 88 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.235458631131937480205166274512, −8.639292781240572158557028757728, −8.141240431057754143222471996132, −7.72197071317775876043770391972, −7.39626741374466874403803585092, −6.59122979173333645814567720322, −6.18137254046148659224375188019, −5.41096095388497296190944805699, −5.13876435477390909749549888363, −4.81747873106571338952541075741, −4.11606560621898389569600803044, −3.57378411990651119346270287511, −2.96818425826653488552578623383, −1.99782096615047986568091979087, −1.35168035449608951000230919000,
1.35168035449608951000230919000, 1.99782096615047986568091979087, 2.96818425826653488552578623383, 3.57378411990651119346270287511, 4.11606560621898389569600803044, 4.81747873106571338952541075741, 5.13876435477390909749549888363, 5.41096095388497296190944805699, 6.18137254046148659224375188019, 6.59122979173333645814567720322, 7.39626741374466874403803585092, 7.72197071317775876043770391972, 8.141240431057754143222471996132, 8.639292781240572158557028757728, 9.235458631131937480205166274512
