| L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 3·9-s − 2·10-s − 4·11-s + 16-s + 2·17-s + 3·18-s + 2·20-s + 4·22-s − 6·25-s + 10·29-s − 32-s − 2·34-s − 3·36-s − 10·37-s − 2·40-s − 4·44-s − 6·45-s + 4·47-s + 6·49-s + 6·50-s − 8·55-s − 10·58-s − 2·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 9-s − 0.632·10-s − 1.20·11-s + 1/4·16-s + 0.485·17-s + 0.707·18-s + 0.447·20-s + 0.852·22-s − 6/5·25-s + 1.85·29-s − 0.176·32-s − 0.342·34-s − 1/2·36-s − 1.64·37-s − 0.316·40-s − 0.603·44-s − 0.894·45-s + 0.583·47-s + 6/7·49-s + 0.848·50-s − 1.07·55-s − 1.31·58-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
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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
−8.532816213622114096654551281017, −8.169314958115618812595298428598, −7.73559972440307903163711231802, −7.21711250154337967288393541385, −6.68155059850002947589758457202, −6.05940964516922099697910222066, −5.79905073177474180187832103866, −5.26787315094652639960258356084, −4.88544323044523213778601090587, −3.94261860012810658825455622412, −3.25182166272888509607356036238, −2.58739387145482707976838456612, −2.25694907809682576965960697336, −1.28168946633399086984930364789, 0,
1.28168946633399086984930364789, 2.25694907809682576965960697336, 2.58739387145482707976838456612, 3.25182166272888509607356036238, 3.94261860012810658825455622412, 4.88544323044523213778601090587, 5.26787315094652639960258356084, 5.79905073177474180187832103866, 6.05940964516922099697910222066, 6.68155059850002947589758457202, 7.21711250154337967288393541385, 7.73559972440307903163711231802, 8.169314958115618812595298428598, 8.532816213622114096654551281017
