L(s) = 1 | − 2-s + 3-s + 4-s + 2·5-s − 6-s − 8-s − 2·9-s − 2·10-s + 12-s + 2·15-s + 16-s + 2·18-s − 5·19-s + 2·20-s − 3·23-s − 24-s + 2·25-s − 5·27-s + 3·29-s − 2·30-s − 32-s − 2·36-s + 5·38-s − 2·40-s + 7·43-s − 4·45-s + 3·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.353·8-s − 2/3·9-s − 0.632·10-s + 0.288·12-s + 0.516·15-s + 1/4·16-s + 0.471·18-s − 1.14·19-s + 0.447·20-s − 0.625·23-s − 0.204·24-s + 2/5·25-s − 0.962·27-s + 0.557·29-s − 0.365·30-s − 0.176·32-s − 1/3·36-s + 0.811·38-s − 0.316·40-s + 1.06·43-s − 0.596·45-s + 0.442·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8084540150\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8084540150\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) |
good | 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 103 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 113 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−11.94879663068125396855491505755, −11.48025403259353863848869431157, −10.76993179663608470720790519061, −10.23892729710416322192518254732, −9.729049277107821360014115576610, −9.121933382765696920995619073396, −8.532832237284381714477548414914, −8.178114840955523972607042414056, −7.33404004829892313575769766146, −6.49078734165607754173981784218, −6.00077677817699022382696369696, −5.18857919569295457660785613666, −4.01289098763868268766009594022, −2.84628359590963579318435033144, −2.00306439532167969849245728601,
2.00306439532167969849245728601, 2.84628359590963579318435033144, 4.01289098763868268766009594022, 5.18857919569295457660785613666, 6.00077677817699022382696369696, 6.49078734165607754173981784218, 7.33404004829892313575769766146, 8.178114840955523972607042414056, 8.532832237284381714477548414914, 9.121933382765696920995619073396, 9.729049277107821360014115576610, 10.23892729710416322192518254732, 10.76993179663608470720790519061, 11.48025403259353863848869431157, 11.94879663068125396855491505755