L(s) = 1 | − 3-s + 7-s + 9-s − 11-s + 5·13-s + 4·17-s − 5·19-s − 21-s + 6·23-s − 27-s − 9·29-s + 4·31-s + 33-s − 5·37-s − 5·39-s − 6·43-s − 7·47-s + 49-s − 4·51-s − 2·53-s + 5·57-s + 3·59-s − 14·61-s + 63-s + 7·67-s − 6·69-s − 9·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.38·13-s + 0.970·17-s − 1.14·19-s − 0.218·21-s + 1.25·23-s − 0.192·27-s − 1.67·29-s + 0.718·31-s + 0.174·33-s − 0.821·37-s − 0.800·39-s − 0.914·43-s − 1.02·47-s + 1/7·49-s − 0.560·51-s − 0.274·53-s + 0.662·57-s + 0.390·59-s − 1.79·61-s + 0.125·63-s + 0.855·67-s − 0.722·69-s − 1.05·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.786092036\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.786092036\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.73530954513709, −14.10783169010177, −13.42395553131342, −13.03310710385706, −12.70807655578894, −11.85159867831061, −11.56045127920580, −10.88504902405275, −10.64888131992275, −10.07483832049217, −9.327093814831924, −8.828531819614447, −8.232270539542653, −7.793248554880192, −7.047038844223419, −6.550855307221960, −5.924352700720163, −5.460908438703389, −4.851039361341574, −4.237792837725387, −3.497236182799481, −3.025778736532488, −1.871904931299051, −1.434915308339267, −0.5100645158817144,
0.5100645158817144, 1.434915308339267, 1.871904931299051, 3.025778736532488, 3.497236182799481, 4.237792837725387, 4.851039361341574, 5.460908438703389, 5.924352700720163, 6.550855307221960, 7.047038844223419, 7.793248554880192, 8.232270539542653, 8.828531819614447, 9.327093814831924, 10.07483832049217, 10.64888131992275, 10.88504902405275, 11.56045127920580, 11.85159867831061, 12.70807655578894, 13.03310710385706, 13.42395553131342, 14.10783169010177, 14.73530954513709