| L(s) = 1 | + 2-s + 4·3-s − 4-s + 4·6-s − 3·8-s + 6·9-s + 2·11-s − 4·12-s − 16-s + 8·17-s + 6·18-s + 2·22-s − 12·24-s − 6·25-s − 4·27-s + 5·32-s + 8·33-s + 8·34-s − 6·36-s + 8·41-s + 24·43-s − 2·44-s − 4·48-s + 49-s − 6·50-s + 32·51-s − 4·54-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 2.30·3-s − 1/2·4-s + 1.63·6-s − 1.06·8-s + 2·9-s + 0.603·11-s − 1.15·12-s − 1/4·16-s + 1.94·17-s + 1.41·18-s + 0.426·22-s − 2.44·24-s − 6/5·25-s − 0.769·27-s + 0.883·32-s + 1.39·33-s + 1.37·34-s − 36-s + 1.24·41-s + 3.65·43-s − 0.301·44-s − 0.577·48-s + 1/7·49-s − 0.848·50-s + 4.48·51-s − 0.544·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.039087435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.039087435\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.678915900547504955372461266927, −8.318055770167249209020787556529, −7.82867506007281568614416512666, −7.50242612035465341765985002833, −7.15095220920519743848333951211, −5.96443978110022033983848919969, −5.89346819607895550605187044438, −5.47251856550031920805970764164, −4.46493777161093979419971313423, −4.07930883034575373640745113861, −3.61703962519106612479648235899, −3.26888344602981049454952325554, −2.61963574407090955942422458621, −2.22212830718058091155401288728, −1.05751961479654851181720694519,
1.05751961479654851181720694519, 2.22212830718058091155401288728, 2.61963574407090955942422458621, 3.26888344602981049454952325554, 3.61703962519106612479648235899, 4.07930883034575373640745113861, 4.46493777161093979419971313423, 5.47251856550031920805970764164, 5.89346819607895550605187044438, 5.96443978110022033983848919969, 7.15095220920519743848333951211, 7.50242612035465341765985002833, 7.82867506007281568614416512666, 8.318055770167249209020787556529, 8.678915900547504955372461266927
