L(s) = 1 | − 3·3-s + 4·4-s + 4·7-s + 6·9-s − 12·12-s − 2·13-s + 12·16-s + 15·19-s − 12·21-s + 5·25-s − 9·27-s + 16·28-s − 15·31-s + 24·36-s − 22·37-s + 6·39-s + 8·43-s − 36·48-s + 9·49-s − 8·52-s − 45·57-s − 27·61-s + 24·63-s + 32·64-s − 11·67-s − 24·73-s − 15·75-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 2·4-s + 1.51·7-s + 2·9-s − 3.46·12-s − 0.554·13-s + 3·16-s + 3.44·19-s − 2.61·21-s + 25-s − 1.73·27-s + 3.02·28-s − 2.69·31-s + 4·36-s − 3.61·37-s + 0.960·39-s + 1.21·43-s − 5.19·48-s + 9/7·49-s − 1.10·52-s − 5.96·57-s − 3.45·61-s + 3.02·63-s + 4·64-s − 1.34·67-s − 2.80·73-s − 1.73·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.738686864\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.738686864\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−11.87147962314324408927532650214, −11.80201489812834484710536665064, −11.05492240262172909463277311833, −11.04352919388191984109753968112, −10.39319952387523364057149836898, −10.34030112497915912392582765464, −9.358892255116999374650434056434, −8.910111944363342481758103982556, −7.75752812144092765319546872157, −7.53654773184354997369529670453, −7.11526932273722673105108455125, −6.99803199730055645006653172379, −5.91618307430411832413316018983, −5.64101106934306220767374995501, −5.18736756205102008593806676804, −4.83483730757439600057775632918, −3.56940680954281740674240294168, −2.96282131225023266566178098366, −1.55479489252007004726710845250, −1.49903082663164853734163490938,
1.49903082663164853734163490938, 1.55479489252007004726710845250, 2.96282131225023266566178098366, 3.56940680954281740674240294168, 4.83483730757439600057775632918, 5.18736756205102008593806676804, 5.64101106934306220767374995501, 5.91618307430411832413316018983, 6.99803199730055645006653172379, 7.11526932273722673105108455125, 7.53654773184354997369529670453, 7.75752812144092765319546872157, 8.910111944363342481758103982556, 9.358892255116999374650434056434, 10.34030112497915912392582765464, 10.39319952387523364057149836898, 11.04352919388191984109753968112, 11.05492240262172909463277311833, 11.80201489812834484710536665064, 11.87147962314324408927532650214