| L(s) = 1 | + 3·3-s + 3·5-s − 7-s + 6·9-s + 3·11-s + 5·13-s + 9·15-s + 2·17-s + 4·19-s − 3·21-s − 4·23-s + 4·25-s + 9·27-s − 7·31-s + 9·33-s − 3·35-s − 2·37-s + 15·39-s − 8·41-s − 43-s + 18·45-s − 3·47-s + 49-s + 6·51-s + 9·53-s + 9·55-s + 12·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.34·5-s − 0.377·7-s + 2·9-s + 0.904·11-s + 1.38·13-s + 2.32·15-s + 0.485·17-s + 0.917·19-s − 0.654·21-s − 0.834·23-s + 4/5·25-s + 1.73·27-s − 1.25·31-s + 1.56·33-s − 0.507·35-s − 0.328·37-s + 2.40·39-s − 1.24·41-s − 0.152·43-s + 2.68·45-s − 0.437·47-s + 1/7·49-s + 0.840·51-s + 1.23·53-s + 1.21·55-s + 1.58·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.552454063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.552454063\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
| 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
−13.85538335593717, −13.60726725519859, −12.89354508952324, −12.78840991512840, −11.86048625464917, −11.42054414645472, −10.63757358554709, −9.943714477803690, −9.864917367725244, −9.349367954423944, −8.746555694999935, −8.583845039330973, −7.920510972463943, −7.282959663162892, −6.682855340929234, −6.353207518140806, −5.512448126660734, −5.265779323131345, −4.085432225142532, −3.661657996348703, −3.411535536502995, −2.579556122720509, −1.980988407698016, −1.551864453197216, −0.9098028933600967,
0.9098028933600967, 1.551864453197216, 1.980988407698016, 2.579556122720509, 3.411535536502995, 3.661657996348703, 4.085432225142532, 5.265779323131345, 5.512448126660734, 6.353207518140806, 6.682855340929234, 7.282959663162892, 7.920510972463943, 8.583845039330973, 8.746555694999935, 9.349367954423944, 9.864917367725244, 9.943714477803690, 10.63757358554709, 11.42054414645472, 11.86048625464917, 12.78840991512840, 12.89354508952324, 13.60726725519859, 13.85538335593717
