L(s) = 1 | − 2-s + 4-s + 3·7-s − 8-s + 5·9-s − 3·14-s + 16-s + 6·17-s − 5·18-s − 3·23-s − 2·25-s + 3·28-s − 9·31-s − 32-s − 6·34-s + 5·36-s + 6·41-s + 3·46-s − 9·47-s − 7·49-s + 2·50-s − 3·56-s + 9·62-s + 15·63-s + 64-s + 6·68-s + 9·71-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.13·7-s − 0.353·8-s + 5/3·9-s − 0.801·14-s + 1/4·16-s + 1.45·17-s − 1.17·18-s − 0.625·23-s − 2/5·25-s + 0.566·28-s − 1.61·31-s − 0.176·32-s − 1.02·34-s + 5/6·36-s + 0.937·41-s + 0.442·46-s − 1.31·47-s − 49-s + 0.282·50-s − 0.400·56-s + 1.14·62-s + 1.88·63-s + 1/8·64-s + 0.727·68-s + 1.06·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.218563100\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.218563100\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 17 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 95 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−10.16300031287998155890483758355, −9.820410053193260115875132340102, −9.521579761496950245059105795495, −8.739859514953923329944753655070, −8.082990079658272742673622220373, −7.79290570331884241877252369599, −7.30344576048429336750405296653, −6.82325941329740596727796488894, −5.96287436632305904458532205845, −5.39822021766368328710423481040, −4.66570739970812415093138866448, −4.03776253921539209178242793473, −3.24955342987286671398593041550, −1.91525974832024812591118024386, −1.38473215609387572114413747220,
1.38473215609387572114413747220, 1.91525974832024812591118024386, 3.24955342987286671398593041550, 4.03776253921539209178242793473, 4.66570739970812415093138866448, 5.39822021766368328710423481040, 5.96287436632305904458532205845, 6.82325941329740596727796488894, 7.30344576048429336750405296653, 7.79290570331884241877252369599, 8.082990079658272742673622220373, 8.739859514953923329944753655070, 9.521579761496950245059105795495, 9.820410053193260115875132340102, 10.16300031287998155890483758355