L(s) = 1 | − 2-s + 2·3-s + 4-s − 5-s − 2·6-s + 2·7-s − 8-s + 9-s + 10-s + 6·11-s + 2·12-s − 2·14-s − 2·15-s + 16-s − 18-s − 20-s + 4·21-s − 6·22-s − 2·24-s + 25-s − 4·27-s + 2·28-s + 2·30-s − 32-s + 12·33-s − 2·35-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.80·11-s + 0.577·12-s − 0.534·14-s − 0.516·15-s + 1/4·16-s − 0.235·18-s − 0.223·20-s + 0.872·21-s − 1.27·22-s − 0.408·24-s + 1/5·25-s − 0.769·27-s + 0.377·28-s + 0.365·30-s − 0.176·32-s + 2.08·33-s − 0.338·35-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.454301533\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.454301533\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_1$ | \( 1 + T \) |
good | 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
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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
−10.03206995033638053454789118892, −9.823375295649081380418590920136, −9.145474696294041331632077665653, −8.761341695645055066667341260395, −8.321722450854969435493752818837, −7.998258853901465580969371367477, −7.28254836118582905874103774127, −6.78923565974098429751523140123, −6.25645156373512622315757491177, −5.32980066926380223263880725199, −4.54010664025825939416315316847, −3.74777133045768309466283974591, −3.32095382498356817710621977137, −2.21911413208596523989656403630, −1.40435653997204247929252733124,
1.40435653997204247929252733124, 2.21911413208596523989656403630, 3.32095382498356817710621977137, 3.74777133045768309466283974591, 4.54010664025825939416315316847, 5.32980066926380223263880725199, 6.25645156373512622315757491177, 6.78923565974098429751523140123, 7.28254836118582905874103774127, 7.998258853901465580969371367477, 8.321722450854969435493752818837, 8.761341695645055066667341260395, 9.145474696294041331632077665653, 9.823375295649081380418590920136, 10.03206995033638053454789118892