| L(s) = 1 | − 2·3-s + 6·7-s − 3·9-s − 6·11-s − 4·16-s − 12·21-s + 6·25-s + 14·27-s + 12·33-s − 2·37-s − 6·41-s + 6·47-s + 8·48-s + 13·49-s + 18·53-s − 18·63-s − 24·67-s − 6·71-s + 18·73-s − 12·75-s − 36·77-s − 4·81-s + 18·83-s + 18·99-s − 6·101-s − 24·107-s + 4·111-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 2.26·7-s − 9-s − 1.80·11-s − 16-s − 2.61·21-s + 6/5·25-s + 2.69·27-s + 2.08·33-s − 0.328·37-s − 0.937·41-s + 0.875·47-s + 1.15·48-s + 13/7·49-s + 2.47·53-s − 2.26·63-s − 2.93·67-s − 0.712·71-s + 2.10·73-s − 1.38·75-s − 4.10·77-s − 4/9·81-s + 1.97·83-s + 1.80·99-s − 0.597·101-s − 2.32·107-s + 0.379·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4468046130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4468046130\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 37 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−16.80724976697154758132500055494, −16.35243031038882138361455797987, −15.61340145096855589033951217458, −14.74467812005020822162815223724, −14.69261999774586902257055533599, −13.66233242554910635758515229827, −13.41483824783508981828592608142, −12.03641000418996993895043545230, −12.00512485502068726851633512346, −11.17371330004469292285283552681, −10.70164083949079417779568466719, −10.57595062939260961208204296645, −8.962523271675309961068729949697, −8.405568605735233663479975827973, −7.87661949667809268185967698094, −6.91264800434440678178571984743, −5.72792428408809935011656816428, −5.12350748025181051471923975855, −4.76215737775444926855823515210, −2.55062267981930742747221241338,
2.55062267981930742747221241338, 4.76215737775444926855823515210, 5.12350748025181051471923975855, 5.72792428408809935011656816428, 6.91264800434440678178571984743, 7.87661949667809268185967698094, 8.405568605735233663479975827973, 8.962523271675309961068729949697, 10.57595062939260961208204296645, 10.70164083949079417779568466719, 11.17371330004469292285283552681, 12.00512485502068726851633512346, 12.03641000418996993895043545230, 13.41483824783508981828592608142, 13.66233242554910635758515229827, 14.69261999774586902257055533599, 14.74467812005020822162815223724, 15.61340145096855589033951217458, 16.35243031038882138361455797987, 16.80724976697154758132500055494
