| L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s + 2·9-s − 2·14-s + 16-s − 4·17-s + 2·18-s + 8·23-s + 25-s − 2·28-s + 8·31-s + 32-s − 4·34-s + 2·36-s + 4·41-s + 8·46-s + 3·49-s + 50-s − 2·56-s + 8·62-s − 4·63-s + 64-s − 4·68-s + 8·71-s + 2·72-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 2/3·9-s − 0.534·14-s + 1/4·16-s − 0.970·17-s + 0.471·18-s + 1.66·23-s + 1/5·25-s − 0.377·28-s + 1.43·31-s + 0.176·32-s − 0.685·34-s + 1/3·36-s + 0.624·41-s + 1.17·46-s + 3/7·49-s + 0.141·50-s − 0.267·56-s + 1.01·62-s − 0.503·63-s + 1/8·64-s − 0.485·68-s + 0.949·71-s + 0.235·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.508476970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.508476970\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−9.390969242198454154411537614299, −8.738956946755145619662635657084, −8.360942150132234994297890118998, −7.65337605808110577885768009171, −7.08432493200540288951654950314, −6.80325129966316000747834841671, −6.35604871974405890622661732715, −5.78974029904300270500720489264, −5.12055967260496804361534114236, −4.56494881346415403926369144671, −4.19931634888053547234877347202, −3.37314801107385035550363478258, −2.87049843313618670610334442196, −2.15222777588374081933951521130, −0.999775914784122715398374131153,
0.999775914784122715398374131153, 2.15222777588374081933951521130, 2.87049843313618670610334442196, 3.37314801107385035550363478258, 4.19931634888053547234877347202, 4.56494881346415403926369144671, 5.12055967260496804361534114236, 5.78974029904300270500720489264, 6.35604871974405890622661732715, 6.80325129966316000747834841671, 7.08432493200540288951654950314, 7.65337605808110577885768009171, 8.360942150132234994297890118998, 8.738956946755145619662635657084, 9.390969242198454154411537614299
