| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 8·11-s − 12-s + 4·13-s + 16-s + 18-s − 8·22-s − 8·23-s − 24-s − 6·25-s + 4·26-s − 27-s + 32-s + 8·33-s + 36-s + 20·37-s − 4·39-s − 8·44-s − 8·46-s − 8·47-s − 48-s − 14·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 2.41·11-s − 0.288·12-s + 1.10·13-s + 1/4·16-s + 0.235·18-s − 1.70·22-s − 1.66·23-s − 0.204·24-s − 6/5·25-s + 0.784·26-s − 0.192·27-s + 0.176·32-s + 1.39·33-s + 1/6·36-s + 3.28·37-s − 0.640·39-s − 1.20·44-s − 1.17·46-s − 1.16·47-s − 0.144·48-s − 2·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 311904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.778956734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.778956734\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.501995439933935386642992901059, −8.057224083383321383561004935430, −7.987570621007900466899230715503, −7.54188937600423423101808663065, −6.59739773494044819386161896201, −6.46925034993373182658971924423, −5.66755609194805589043695166783, −5.60441811878021212455097225817, −5.02382748416547556621669886282, −4.39790054298019034493919320980, −3.86301997176052042051145556459, −3.33052988139656335920774502455, −2.33973473634575001863443152397, −2.15606001082909168122252072981, −0.69408070740984200675842868720,
0.69408070740984200675842868720, 2.15606001082909168122252072981, 2.33973473634575001863443152397, 3.33052988139656335920774502455, 3.86301997176052042051145556459, 4.39790054298019034493919320980, 5.02382748416547556621669886282, 5.60441811878021212455097225817, 5.66755609194805589043695166783, 6.46925034993373182658971924423, 6.59739773494044819386161896201, 7.54188937600423423101808663065, 7.987570621007900466899230715503, 8.057224083383321383561004935430, 8.501995439933935386642992901059
