L(s) = 1 | + 8·7-s + 9-s − 6·17-s + 8·23-s − 8·31-s + 10·41-s + 16·47-s + 34·49-s + 8·63-s + 16·71-s + 14·73-s − 8·79-s − 8·81-s + 2·89-s − 4·97-s + 16·103-s − 30·113-s − 48·119-s − 15·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 6·153-s + 157-s + ⋯ |
L(s) = 1 | + 3.02·7-s + 1/3·9-s − 1.45·17-s + 1.66·23-s − 1.43·31-s + 1.56·41-s + 2.33·47-s + 34/7·49-s + 1.00·63-s + 1.89·71-s + 1.63·73-s − 0.900·79-s − 8/9·81-s + 0.211·89-s − 0.406·97-s + 1.57·103-s − 2.82·113-s − 4.40·119-s − 1.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.485·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.569555243\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.569555243\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 71 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 103 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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
−8.162475697531223146269450891657, −7.82850657812127283405008972941, −7.51912797924572778889452809468, −7.27777570293169737795473330128, −6.88697473407551430949172026276, −6.60519840329478162437117870180, −5.93677057846133888983778425086, −5.57910313558857586874057516929, −5.29422744765153280104290902466, −4.88309594770335573333794249793, −4.71870388229105096542154952376, −4.29431294743316650219918917896, −3.92680933686811581977002060459, −3.65316871584912584481991525708, −2.59315746224210677790079331192, −2.57435859360292555237855641968, −2.00486353130738188885043977501, −1.59662718783930337418300270791, −1.14698927854776593189027082004, −0.63739428570904934513409823509,
0.63739428570904934513409823509, 1.14698927854776593189027082004, 1.59662718783930337418300270791, 2.00486353130738188885043977501, 2.57435859360292555237855641968, 2.59315746224210677790079331192, 3.65316871584912584481991525708, 3.92680933686811581977002060459, 4.29431294743316650219918917896, 4.71870388229105096542154952376, 4.88309594770335573333794249793, 5.29422744765153280104290902466, 5.57910313558857586874057516929, 5.93677057846133888983778425086, 6.60519840329478162437117870180, 6.88697473407551430949172026276, 7.27777570293169737795473330128, 7.51912797924572778889452809468, 7.82850657812127283405008972941, 8.162475697531223146269450891657