| L(s) = 1 | − 0.806·3-s − 2.15·7-s − 2.35·9-s + 0.387·11-s − 0.962·13-s + 1.61·17-s − 6.31·19-s + 1.73·21-s + 6.15·23-s + 4.31·27-s − 2·29-s − 9.92·31-s − 0.312·33-s + 6.57·37-s + 0.775·39-s + 4.57·41-s − 11.5·43-s + 4.54·47-s − 2.35·49-s − 1.29·51-s + 8.96·53-s + 5.08·57-s + 6.31·59-s − 0.261·61-s + 5.06·63-s − 9.89·67-s − 4.96·69-s + ⋯ |
| L(s) = 1 | − 0.465·3-s − 0.815·7-s − 0.783·9-s + 0.116·11-s − 0.266·13-s + 0.390·17-s − 1.44·19-s + 0.379·21-s + 1.28·23-s + 0.829·27-s − 0.371·29-s − 1.78·31-s − 0.0544·33-s + 1.08·37-s + 0.124·39-s + 0.714·41-s − 1.75·43-s + 0.662·47-s − 0.335·49-s − 0.181·51-s + 1.23·53-s + 0.673·57-s + 0.821·59-s − 0.0335·61-s + 0.638·63-s − 1.20·67-s − 0.597·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9168362138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9168362138\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 0.806T + 3T^{2} \) |
| 7 | \( 1 + 2.15T + 7T^{2} \) |
| 11 | \( 1 - 0.387T + 11T^{2} \) |
| 13 | \( 1 + 0.962T + 13T^{2} \) |
| 17 | \( 1 - 1.61T + 17T^{2} \) |
| 19 | \( 1 + 6.31T + 19T^{2} \) |
| 23 | \( 1 - 6.15T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 9.92T + 31T^{2} \) |
| 37 | \( 1 - 6.57T + 37T^{2} \) |
| 41 | \( 1 - 4.57T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 - 4.54T + 47T^{2} \) |
| 53 | \( 1 - 8.96T + 53T^{2} \) |
| 59 | \( 1 - 6.31T + 59T^{2} \) |
| 61 | \( 1 + 0.261T + 61T^{2} \) |
| 67 | \( 1 + 9.89T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 - 1.92T + 79T^{2} \) |
| 83 | \( 1 + 2.88T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 9.61T + 97T^{2} \) |
| 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
−8.814450036320207236867188166001, −7.920254475087629462228722132087, −6.99794648745079350428838989429, −6.40881341252020935093690628261, −5.66634433935500030112312932891, −4.97811487647161179028334933791, −3.90226840973219944197339389158, −3.08900828706616228343418263650, −2.12314852338868049838079747333, −0.56364536844190097919749159071,
0.56364536844190097919749159071, 2.12314852338868049838079747333, 3.08900828706616228343418263650, 3.90226840973219944197339389158, 4.97811487647161179028334933791, 5.66634433935500030112312932891, 6.40881341252020935093690628261, 6.99794648745079350428838989429, 7.920254475087629462228722132087, 8.814450036320207236867188166001
