| L(s) = 1 | − 2·2-s − 4·3-s + 4-s + 2·5-s + 8·6-s − 4·7-s + 6·9-s − 4·10-s − 4·11-s − 4·12-s − 4·13-s + 8·14-s − 8·15-s + 16-s − 12·18-s − 4·19-s + 2·20-s + 16·21-s + 8·22-s − 12·23-s + 3·25-s + 8·26-s + 4·27-s − 4·28-s + 2·29-s + 16·30-s − 4·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 1/2·4-s + 0.894·5-s + 3.26·6-s − 1.51·7-s + 2·9-s − 1.26·10-s − 1.20·11-s − 1.15·12-s − 1.10·13-s + 2.13·14-s − 2.06·15-s + 1/4·16-s − 2.82·18-s − 0.917·19-s + 0.447·20-s + 3.49·21-s + 1.70·22-s − 2.50·23-s + 3/5·25-s + 1.56·26-s + 0.769·27-s − 0.755·28-s + 0.371·29-s + 2.92·30-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 29 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_4$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 4 T - 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 94 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 20 T + 258 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 138 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−12.56082033727498085085278842002, −12.09330472206335963966379431081, −11.94277012482732955786275470974, −11.17398498211377678449527749359, −10.45858470814087163645578370747, −10.24832448728220624571135564090, −9.835621547290741269783234328837, −9.707875196311652840487049333439, −8.595769162577853354147285851763, −8.399942220823804521006680538118, −7.46035497824188566088516866615, −6.63477579800586799219060691200, −6.23865594298435037206375773972, −5.99887355317936411790296248894, −5.11685863390931392049616797616, −4.85844984552289833623841604785, −3.32970587142951762161143985525, −2.17766922491229158005785092950, 0, 0,
2.17766922491229158005785092950, 3.32970587142951762161143985525, 4.85844984552289833623841604785, 5.11685863390931392049616797616, 5.99887355317936411790296248894, 6.23865594298435037206375773972, 6.63477579800586799219060691200, 7.46035497824188566088516866615, 8.399942220823804521006680538118, 8.595769162577853354147285851763, 9.707875196311652840487049333439, 9.835621547290741269783234328837, 10.24832448728220624571135564090, 10.45858470814087163645578370747, 11.17398498211377678449527749359, 11.94277012482732955786275470974, 12.09330472206335963966379431081, 12.56082033727498085085278842002
