| L(s) = 1 | − 4·4-s − 2·7-s − 5·9-s + 6·11-s − 8·13-s + 12·16-s + 12·17-s − 10·25-s + 8·28-s − 12·29-s + 20·36-s + 2·37-s + 16·43-s − 24·44-s + 6·47-s − 11·49-s + 32·52-s − 3·53-s + 24·59-s + 10·63-s − 32·64-s − 48·68-s − 12·77-s + 16·81-s + 12·89-s + 16·91-s + 16·97-s + ⋯ |
| L(s) = 1 | − 2·4-s − 0.755·7-s − 5/3·9-s + 1.80·11-s − 2.21·13-s + 3·16-s + 2.91·17-s − 2·25-s + 1.51·28-s − 2.22·29-s + 10/3·36-s + 0.328·37-s + 2.43·43-s − 3.61·44-s + 0.875·47-s − 1.57·49-s + 4.43·52-s − 0.412·53-s + 3.12·59-s + 1.25·63-s − 4·64-s − 5.82·68-s − 1.36·77-s + 16/9·81-s + 1.27·89-s + 1.67·91-s + 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3845521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3845521 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6510227324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6510227324\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 53 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−7.59911177067371416247669368567, −7.35695888616381028359115098282, −6.39230647045720975486313841427, −6.03020856068344384971964909281, −5.58909751562265465393513577533, −5.44973416215471530225317007593, −5.11578085003588418364185044479, −4.36860820173470907739073935244, −3.94512506936774143358592547197, −3.50910294340479626549928321873, −3.42363177153759690827715424028, −2.64353356431080463432221163659, −1.94027095266097412587366859555, −0.948641580146169613807507685106, −0.38803840635058025951831939568,
0.38803840635058025951831939568, 0.948641580146169613807507685106, 1.94027095266097412587366859555, 2.64353356431080463432221163659, 3.42363177153759690827715424028, 3.50910294340479626549928321873, 3.94512506936774143358592547197, 4.36860820173470907739073935244, 5.11578085003588418364185044479, 5.44973416215471530225317007593, 5.58909751562265465393513577533, 6.03020856068344384971964909281, 6.39230647045720975486313841427, 7.35695888616381028359115098282, 7.59911177067371416247669368567
