| L(s) = 1 | + 3-s + 4·7-s + 9-s + 4·11-s + 4·13-s − 2·17-s + 4·21-s + 8·23-s − 5·25-s + 27-s − 8·29-s − 4·31-s + 4·33-s − 4·37-s + 4·39-s − 6·41-s + 4·43-s + 8·47-s + 9·49-s − 2·51-s − 8·53-s + 12·59-s − 12·61-s + 4·63-s − 12·67-s + 8·69-s + 8·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s + 1.20·11-s + 1.10·13-s − 0.485·17-s + 0.872·21-s + 1.66·23-s − 25-s + 0.192·27-s − 1.48·29-s − 0.718·31-s + 0.696·33-s − 0.657·37-s + 0.640·39-s − 0.937·41-s + 0.609·43-s + 1.16·47-s + 9/7·49-s − 0.280·51-s − 1.09·53-s + 1.56·59-s − 1.53·61-s + 0.503·63-s − 1.46·67-s + 0.963·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.665647651\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.665647651\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.86569416009618, −12.22259232227594, −11.75773340635083, −11.27425119080885, −11.03660395629445, −10.65577179442125, −9.956101694546623, −9.200543279368042, −9.083786760627470, −8.632054177578185, −8.238442599944072, −7.517246841148061, −7.304943317192642, −6.768903324494763, −6.049417435948066, −5.670473391370517, −5.050157362677421, −4.483962749377612, −4.076914378819987, −3.540980235488512, −3.108545442357180, −2.136666160721966, −1.713347407435079, −1.387953271945893, −0.6163724415038104,
0.6163724415038104, 1.387953271945893, 1.713347407435079, 2.136666160721966, 3.108545442357180, 3.540980235488512, 4.076914378819987, 4.483962749377612, 5.050157362677421, 5.670473391370517, 6.049417435948066, 6.768903324494763, 7.304943317192642, 7.517246841148061, 8.238442599944072, 8.632054177578185, 9.083786760627470, 9.200543279368042, 9.956101694546623, 10.65577179442125, 11.03660395629445, 11.27425119080885, 11.75773340635083, 12.22259232227594, 12.86569416009618
