| L(s) = 1 | − 9-s + 8·13-s − 14·17-s + 4·37-s + 10·41-s + 6·49-s + 12·53-s + 20·61-s − 18·73-s − 8·81-s − 10·89-s + 4·97-s − 4·101-s + 12·109-s − 2·113-s − 8·117-s − 17·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 14·153-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 2.21·13-s − 3.39·17-s + 0.657·37-s + 1.56·41-s + 6/7·49-s + 1.64·53-s + 2.56·61-s − 2.10·73-s − 8/9·81-s − 1.05·89-s + 0.406·97-s − 0.398·101-s + 1.14·109-s − 0.188·113-s − 0.739·117-s − 1.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.13·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.724888820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.724888820\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 129 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 138 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−10.50605509965111304486465515300, −10.24993019668767040986526642586, −9.464476774997252509084814202742, −9.014783488310806066676906286092, −8.826822029803336769437210452676, −8.465430342484352186093050379282, −8.167905739342344370023067006366, −7.34755154133375962825895060082, −6.94760004129955496294405041358, −6.59803878243940616364742111278, −5.94426147354046117803981663122, −5.93614136368984664489504438907, −5.15500699103202102429843650560, −4.43230042203715345727770067725, −4.01699601747083075472287508810, −3.86599210376161220926146517499, −2.76239025809467826674546430899, −2.43843119291001336407519068539, −1.64894313034439815950593792217, −0.66921415445912516203127009829,
0.66921415445912516203127009829, 1.64894313034439815950593792217, 2.43843119291001336407519068539, 2.76239025809467826674546430899, 3.86599210376161220926146517499, 4.01699601747083075472287508810, 4.43230042203715345727770067725, 5.15500699103202102429843650560, 5.93614136368984664489504438907, 5.94426147354046117803981663122, 6.59803878243940616364742111278, 6.94760004129955496294405041358, 7.34755154133375962825895060082, 8.167905739342344370023067006366, 8.465430342484352186093050379282, 8.826822029803336769437210452676, 9.014783488310806066676906286092, 9.464476774997252509084814202742, 10.24993019668767040986526642586, 10.50605509965111304486465515300
