| L(s) = 1 | + 2-s + 5·3-s + 8·4-s − 5·5-s + 5·6-s + 44·7-s + 23·8-s + 27·9-s − 5·10-s + 18·11-s + 40·12-s − 54·13-s + 44·14-s − 25·15-s + 23·16-s + 54·17-s + 27·18-s − 133·19-s − 40·20-s + 220·21-s + 18·22-s + 92·23-s + 115·24-s − 54·26-s + 280·27-s + 352·28-s + 134·29-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 0.962·3-s + 4-s − 0.447·5-s + 0.340·6-s + 2.37·7-s + 1.01·8-s + 9-s − 0.158·10-s + 0.493·11-s + 0.962·12-s − 1.15·13-s + 0.839·14-s − 0.430·15-s + 0.359·16-s + 0.770·17-s + 0.353·18-s − 1.60·19-s − 0.447·20-s + 2.28·21-s + 0.174·22-s + 0.834·23-s + 0.978·24-s − 0.407·26-s + 1.99·27-s + 2.37·28-s + 0.858·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.322927915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.322927915\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 19 | $C_2$ | \( 1 + 7 p T + p^{3} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - 7 T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 5 T - 2 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 22 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 9 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 54 T + 719 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 54 T - 1997 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 p T - 7 p^{2} T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 134 T - 6433 T^{2} - 134 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 252 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 236 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 243 T - 9872 T^{2} - 243 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 496 T + 166509 T^{2} + 496 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 502 T + 148181 T^{2} + 502 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 62 T - 145033 T^{2} + 62 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 681 T + 258382 T^{2} + 681 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 142 T - 206817 T^{2} - 142 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 55 T - 297738 T^{2} + 55 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 974 T + 590765 T^{2} - 974 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 695 T + 94008 T^{2} + 695 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 736 T + 48657 T^{2} - 736 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 63 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 726 T - 177893 T^{2} + 726 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 1167 T + 449216 T^{2} - 1167 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−14.21712530343059068595670031907, −13.27845319861924151200689279822, −12.44058131781760656095872224115, −12.42966042577659884862406238900, −11.44486107416496694412240750897, −11.27649951212410368447791092906, −10.56107768258407645578290151821, −10.31272078955377143794714183235, −9.218415360922711377934697209765, −8.604153506381280619778995602243, −8.079984804409519744365686719613, −7.65583369751070102026145787483, −7.06685413320404984814767140528, −6.62523077144777768688396095426, −5.05702198807915921117737565279, −4.92045739555757377399174611867, −4.13421292185354867165874738185, −3.13095906600118597961816098731, −1.81864937723505996313741253251, −1.74699357876445148617334030237,
1.74699357876445148617334030237, 1.81864937723505996313741253251, 3.13095906600118597961816098731, 4.13421292185354867165874738185, 4.92045739555757377399174611867, 5.05702198807915921117737565279, 6.62523077144777768688396095426, 7.06685413320404984814767140528, 7.65583369751070102026145787483, 8.079984804409519744365686719613, 8.604153506381280619778995602243, 9.218415360922711377934697209765, 10.31272078955377143794714183235, 10.56107768258407645578290151821, 11.27649951212410368447791092906, 11.44486107416496694412240750897, 12.42966042577659884862406238900, 12.44058131781760656095872224115, 13.27845319861924151200689279822, 14.21712530343059068595670031907
