| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 8-s − 3·9-s + 2·11-s + 2·12-s + 16-s + 6·17-s + 3·18-s − 14·19-s − 2·22-s − 2·24-s + 25-s − 14·27-s − 32-s + 4·33-s − 6·34-s − 3·36-s + 14·38-s + 12·41-s + 16·43-s + 2·44-s + 2·48-s + 11·49-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.353·8-s − 9-s + 0.603·11-s + 0.577·12-s + 1/4·16-s + 1.45·17-s + 0.707·18-s − 3.21·19-s − 0.426·22-s − 0.408·24-s + 1/5·25-s − 2.69·27-s − 0.176·32-s + 0.696·33-s − 1.02·34-s − 1/2·36-s + 2.27·38-s + 1.87·41-s + 2.43·43-s + 0.301·44-s + 0.288·48-s + 11/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.554192200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.554192200\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
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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
−8.745336918010044944273252806376, −8.285662434899332908926558240119, −7.909333761467571091663115471270, −7.56981196469027501153188760806, −6.96458110048007349200287196794, −6.29237276205882136661088933535, −5.85902806217107753681965596133, −5.73876439085829035805370640147, −4.67797946529805246414945801346, −4.01173135747517138761270457828, −3.69942049216283911588153612758, −2.85810437100354775911014337424, −2.44066996588343376455793034646, −1.97170842587956421114470880781, −0.70426495877526572877052419514,
0.70426495877526572877052419514, 1.97170842587956421114470880781, 2.44066996588343376455793034646, 2.85810437100354775911014337424, 3.69942049216283911588153612758, 4.01173135747517138761270457828, 4.67797946529805246414945801346, 5.73876439085829035805370640147, 5.85902806217107753681965596133, 6.29237276205882136661088933535, 6.96458110048007349200287196794, 7.56981196469027501153188760806, 7.909333761467571091663115471270, 8.285662434899332908926558240119, 8.745336918010044944273252806376
