| L(s) = 1 | − 2·5-s − 7-s + 3·9-s − 2·11-s + 7·13-s + 3·17-s − 6·19-s − 4·23-s − 7·25-s + 7·29-s − 8·31-s + 2·35-s − 9·37-s − 9·41-s + 10·43-s − 6·45-s + 4·47-s + 18·53-s + 4·55-s + 14·59-s + 5·61-s − 3·63-s − 14·65-s − 8·67-s + 10·71-s − 14·73-s + 2·77-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s + 9-s − 0.603·11-s + 1.94·13-s + 0.727·17-s − 1.37·19-s − 0.834·23-s − 7/5·25-s + 1.29·29-s − 1.43·31-s + 0.338·35-s − 1.47·37-s − 1.40·41-s + 1.52·43-s − 0.894·45-s + 0.583·47-s + 2.47·53-s + 0.539·55-s + 1.82·59-s + 0.640·61-s − 0.377·63-s − 1.73·65-s − 0.977·67-s + 1.18·71-s − 1.63·73-s + 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2119936 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2119936 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.416877339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.416877339\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 9 T + 44 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 10 T + 29 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.957092576590929367851818577979, −9.114057566145506150334802496250, −8.784912819514252520236630864563, −8.629262595615513338256919224167, −8.063469087569654328525366741188, −7.74811084883029840071587525329, −7.38559778192496063108922249683, −6.85669009648069150113453610028, −6.55317747396699365239383293319, −6.02302168340223156310534432405, −5.59314324856627679174299047848, −5.28790517537526930567765011410, −4.47295037722606286586029578956, −3.94016288269780118022406740548, −3.82107033005566794003713054272, −3.55648629853055582686055585539, −2.61643743439669012360120528646, −2.01801590895761174109505791876, −1.40151883700983408992146203129, −0.49870685824528081504784984928,
0.49870685824528081504784984928, 1.40151883700983408992146203129, 2.01801590895761174109505791876, 2.61643743439669012360120528646, 3.55648629853055582686055585539, 3.82107033005566794003713054272, 3.94016288269780118022406740548, 4.47295037722606286586029578956, 5.28790517537526930567765011410, 5.59314324856627679174299047848, 6.02302168340223156310534432405, 6.55317747396699365239383293319, 6.85669009648069150113453610028, 7.38559778192496063108922249683, 7.74811084883029840071587525329, 8.063469087569654328525366741188, 8.629262595615513338256919224167, 8.784912819514252520236630864563, 9.114057566145506150334802496250, 9.957092576590929367851818577979
