| L(s) = 1 | + 2-s − 2·5-s + 2·7-s + 8-s − 2·10-s − 2·11-s + 6·13-s + 2·14-s − 16-s + 4·17-s − 2·22-s + 6·23-s + 3·25-s + 6·26-s − 10·29-s − 4·31-s − 6·32-s + 4·34-s − 4·35-s + 4·37-s − 2·40-s + 2·41-s − 6·43-s + 6·46-s + 4·47-s + 2·49-s + 3·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.894·5-s + 0.755·7-s + 0.353·8-s − 0.632·10-s − 0.603·11-s + 1.66·13-s + 0.534·14-s − 1/4·16-s + 0.970·17-s − 0.426·22-s + 1.25·23-s + 3/5·25-s + 1.17·26-s − 1.85·29-s − 0.718·31-s − 1.06·32-s + 0.685·34-s − 0.676·35-s + 0.657·37-s − 0.316·40-s + 0.312·41-s − 0.914·43-s + 0.884·46-s + 0.583·47-s + 2/7·49-s + 0.424·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.511025488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.511025488\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 53 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 97 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 79 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 146 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 22 T + 250 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 18 T + 214 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 209 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−13.26651845725768254820666044922, −13.15656757009923404443434320868, −12.69618351674111987651486792330, −11.86950036690744818610321139579, −11.39992563072627505778202264586, −11.13649202637338755926750469179, −10.65577248819775525810677457588, −10.07946073498264653660591120770, −9.022283179608440638801813663273, −8.851300688689427455059761656759, −8.108019010147471591376132854846, −7.51047393996337455135390379093, −7.25188465063029580597412460768, −6.29145426566632754277559327485, −5.35566276060651074095151355006, −5.24881298504736273695468213845, −4.10459038170170635162919227844, −3.90240385673282395168019440799, −2.93848228502830912393814507342, −1.48047827314022114275690514485,
1.48047827314022114275690514485, 2.93848228502830912393814507342, 3.90240385673282395168019440799, 4.10459038170170635162919227844, 5.24881298504736273695468213845, 5.35566276060651074095151355006, 6.29145426566632754277559327485, 7.25188465063029580597412460768, 7.51047393996337455135390379093, 8.108019010147471591376132854846, 8.851300688689427455059761656759, 9.022283179608440638801813663273, 10.07946073498264653660591120770, 10.65577248819775525810677457588, 11.13649202637338755926750469179, 11.39992563072627505778202264586, 11.86950036690744818610321139579, 12.69618351674111987651486792330, 13.15656757009923404443434320868, 13.26651845725768254820666044922
