| L(s) = 1 | + 2·2-s + 2·3-s + 2·4-s + 4·6-s − 3·9-s − 6·11-s + 4·12-s − 4·16-s + 14·17-s − 6·18-s − 12·22-s − 14·27-s − 8·32-s − 12·33-s + 28·34-s − 6·36-s + 4·41-s − 8·43-s − 12·44-s − 8·48-s + 49-s + 28·51-s − 28·54-s + 20·59-s − 8·64-s − 24·66-s + 4·67-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 4-s + 1.63·6-s − 9-s − 1.80·11-s + 1.15·12-s − 16-s + 3.39·17-s − 1.41·18-s − 2.55·22-s − 2.69·27-s − 1.41·32-s − 2.08·33-s + 4.80·34-s − 36-s + 0.624·41-s − 1.21·43-s − 1.80·44-s − 1.15·48-s + 1/7·49-s + 3.92·51-s − 3.81·54-s + 2.60·59-s − 64-s − 2.95·66-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.075790943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.075790943\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
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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
−7.83592762513447774723326132090, −7.38451703018051703490900915946, −7.05100957889080392304434332168, −6.10912051523666843643835490963, −5.93898740434763322465740513982, −5.45922076877847354484924950710, −5.14936161735373762545181523253, −5.01570061778941535708309486660, −3.95749147690713051772963085752, −3.57752582126466520607951640819, −3.30255970627597751194469732923, −2.76472422088648867371864611007, −2.57959600757021446327298173729, −1.87226019490551330647832359751, −0.64173349430317986722987428715,
0.64173349430317986722987428715, 1.87226019490551330647832359751, 2.57959600757021446327298173729, 2.76472422088648867371864611007, 3.30255970627597751194469732923, 3.57752582126466520607951640819, 3.95749147690713051772963085752, 5.01570061778941535708309486660, 5.14936161735373762545181523253, 5.45922076877847354484924950710, 5.93898740434763322465740513982, 6.10912051523666843643835490963, 7.05100957889080392304434332168, 7.38451703018051703490900915946, 7.83592762513447774723326132090
