| L(s) = 1 | − 3-s + 9-s + 5·11-s + 5·13-s + 4·17-s + 7·19-s + 23-s − 27-s + 2·31-s − 5·33-s − 37-s − 5·39-s + 5·41-s + 12·43-s − 11·47-s − 4·51-s + 9·53-s − 7·57-s − 4·59-s + 4·61-s − 12·67-s − 69-s − 2·71-s − 10·73-s + 12·79-s + 81-s − 12·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.50·11-s + 1.38·13-s + 0.970·17-s + 1.60·19-s + 0.208·23-s − 0.192·27-s + 0.359·31-s − 0.870·33-s − 0.164·37-s − 0.800·39-s + 0.780·41-s + 1.82·43-s − 1.60·47-s − 0.560·51-s + 1.23·53-s − 0.927·57-s − 0.520·59-s + 0.512·61-s − 1.46·67-s − 0.120·69-s − 0.237·71-s − 1.17·73-s + 1.35·79-s + 1/9·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.429996159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.429996159\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.29570989466323, −13.86847774484050, −13.35712941183483, −12.79355619366867, −12.05925777724278, −11.88437195833119, −11.31782535705954, −10.93460660831908, −10.20237509802043, −9.774284026361505, −9.102234352101638, −8.863005375284201, −8.034265015768448, −7.455018440926582, −7.019056076841765, −6.211952077195522, −5.987089267749189, −5.416547824065918, −4.647861086368097, −4.052839712732544, −3.469160989584455, −3.008863785371009, −1.828296599667912, −1.110531831305731, −0.8354237617413224,
0.8354237617413224, 1.110531831305731, 1.828296599667912, 3.008863785371009, 3.469160989584455, 4.052839712732544, 4.647861086368097, 5.416547824065918, 5.987089267749189, 6.211952077195522, 7.019056076841765, 7.455018440926582, 8.034265015768448, 8.863005375284201, 9.102234352101638, 9.774284026361505, 10.20237509802043, 10.93460660831908, 11.31782535705954, 11.88437195833119, 12.05925777724278, 12.79355619366867, 13.35712941183483, 13.86847774484050, 14.29570989466323
