| L(s) = 1 | + 3-s + 7-s − 9-s + 2·11-s − 4·13-s − 3·17-s + 19-s + 21-s − 2·23-s + 7·29-s + 9·31-s + 2·33-s − 37-s − 4·39-s + 4·41-s − 4·43-s + 18·47-s − 9·49-s − 3·51-s − 5·53-s + 57-s − 6·59-s + 17·61-s − 63-s + 4·67-s − 2·69-s − 5·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s − 1/3·9-s + 0.603·11-s − 1.10·13-s − 0.727·17-s + 0.229·19-s + 0.218·21-s − 0.417·23-s + 1.29·29-s + 1.61·31-s + 0.348·33-s − 0.164·37-s − 0.640·39-s + 0.624·41-s − 0.609·43-s + 2.62·47-s − 9/7·49-s − 0.420·51-s − 0.686·53-s + 0.132·57-s − 0.781·59-s + 2.17·61-s − 0.125·63-s + 0.488·67-s − 0.240·69-s − 0.593·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.805651816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.805651816\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T + 34 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 9 T + 78 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T + 36 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 18 T + 158 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 108 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 17 T + 156 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 110 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 14 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 150 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 21 T + 284 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 186 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−9.090934896184569640899879287283, −8.836079158607887779224565736947, −8.396065405275050000548643528401, −8.277890687848119364642440293140, −7.79273216040912319576541389314, −7.25016331243774121086338776401, −6.99783762327816579255765037909, −6.64650717296543303566789539158, −5.95302942267949344024573551505, −5.93313271779471899967796414316, −5.14834217341315200434670866224, −4.77530979654130100975131491836, −4.34319836577439589485487713014, −4.16173793768602393941105862790, −3.24568835410817134030963118390, −3.05815897055407226429013720929, −2.34451272688180751625779055038, −2.18419900016045942772268244648, −1.27951995429994469216233509292, −0.59028211128143221711936960648,
0.59028211128143221711936960648, 1.27951995429994469216233509292, 2.18419900016045942772268244648, 2.34451272688180751625779055038, 3.05815897055407226429013720929, 3.24568835410817134030963118390, 4.16173793768602393941105862790, 4.34319836577439589485487713014, 4.77530979654130100975131491836, 5.14834217341315200434670866224, 5.93313271779471899967796414316, 5.95302942267949344024573551505, 6.64650717296543303566789539158, 6.99783762327816579255765037909, 7.25016331243774121086338776401, 7.79273216040912319576541389314, 8.277890687848119364642440293140, 8.396065405275050000548643528401, 8.836079158607887779224565736947, 9.090934896184569640899879287283
