| L(s) = 1 | + 3·3-s − 24·7-s + 9·9-s − 52·11-s − 22·13-s + 14·17-s + 20·19-s − 72·21-s − 168·23-s + 27·27-s + 230·29-s + 288·31-s − 156·33-s + 34·37-s − 66·39-s + 122·41-s − 188·43-s + 256·47-s + 233·49-s + 42·51-s + 338·53-s + 60·57-s − 100·59-s + 742·61-s − 216·63-s − 84·67-s − 504·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.29·7-s + 1/3·9-s − 1.42·11-s − 0.469·13-s + 0.199·17-s + 0.241·19-s − 0.748·21-s − 1.52·23-s + 0.192·27-s + 1.47·29-s + 1.66·31-s − 0.822·33-s + 0.151·37-s − 0.270·39-s + 0.464·41-s − 0.666·43-s + 0.794·47-s + 0.679·49-s + 0.115·51-s + 0.875·53-s + 0.139·57-s − 0.220·59-s + 1.55·61-s − 0.431·63-s − 0.153·67-s − 0.879·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.604702242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.604702242\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 24 T + p^{3} T^{2} \) |
| 11 | \( 1 + 52 T + p^{3} T^{2} \) |
| 13 | \( 1 + 22 T + p^{3} T^{2} \) |
| 17 | \( 1 - 14 T + p^{3} T^{2} \) |
| 19 | \( 1 - 20 T + p^{3} T^{2} \) |
| 23 | \( 1 + 168 T + p^{3} T^{2} \) |
| 29 | \( 1 - 230 T + p^{3} T^{2} \) |
| 31 | \( 1 - 288 T + p^{3} T^{2} \) |
| 37 | \( 1 - 34 T + p^{3} T^{2} \) |
| 41 | \( 1 - 122 T + p^{3} T^{2} \) |
| 43 | \( 1 + 188 T + p^{3} T^{2} \) |
| 47 | \( 1 - 256 T + p^{3} T^{2} \) |
| 53 | \( 1 - 338 T + p^{3} T^{2} \) |
| 59 | \( 1 + 100 T + p^{3} T^{2} \) |
| 61 | \( 1 - 742 T + p^{3} T^{2} \) |
| 67 | \( 1 + 84 T + p^{3} T^{2} \) |
| 71 | \( 1 - 328 T + p^{3} T^{2} \) |
| 73 | \( 1 - 38 T + p^{3} T^{2} \) |
| 79 | \( 1 - 240 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1212 T + p^{3} T^{2} \) |
| 89 | \( 1 - 330 T + p^{3} T^{2} \) |
| 97 | \( 1 + 866 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.602492183453511830687556715555, −8.402182052767639526673247945800, −7.897010248544882560928834500628, −6.90646913518264071863898902688, −6.11793225168471396021182150610, −5.10376963922713211278983885989, −4.02877163897539893506530420651, −2.95272032675740734678901396032, −2.38542633371338100688012370741, −0.59998993694040901046296524976,
0.59998993694040901046296524976, 2.38542633371338100688012370741, 2.95272032675740734678901396032, 4.02877163897539893506530420651, 5.10376963922713211278983885989, 6.11793225168471396021182150610, 6.90646913518264071863898902688, 7.897010248544882560928834500628, 8.402182052767639526673247945800, 9.602492183453511830687556715555
