| L(s) = 1 | + 8·3-s + 10·5-s − 16·7-s + 37·9-s − 40·11-s + 50·13-s + 80·15-s − 30·17-s + 40·19-s − 128·21-s − 48·23-s − 25·25-s + 80·27-s + 34·29-s − 320·31-s − 320·33-s − 160·35-s − 310·37-s + 400·39-s + 410·41-s + 152·43-s + 370·45-s + 416·47-s − 87·49-s − 240·51-s + 410·53-s − 400·55-s + ⋯ |
| L(s) = 1 | + 1.53·3-s + 0.894·5-s − 0.863·7-s + 1.37·9-s − 1.09·11-s + 1.06·13-s + 1.37·15-s − 0.428·17-s + 0.482·19-s − 1.33·21-s − 0.435·23-s − 1/5·25-s + 0.570·27-s + 0.217·29-s − 1.85·31-s − 1.68·33-s − 0.772·35-s − 1.37·37-s + 1.64·39-s + 1.56·41-s + 0.539·43-s + 1.22·45-s + 1.29·47-s − 0.253·49-s − 0.658·51-s + 1.06·53-s − 0.980·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.259312547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.259312547\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 8 T + p^{3} T^{2} \) |
| 5 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 7 | \( 1 + 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 40 T + p^{3} T^{2} \) |
| 13 | \( 1 - 50 T + p^{3} T^{2} \) |
| 17 | \( 1 + 30 T + p^{3} T^{2} \) |
| 19 | \( 1 - 40 T + p^{3} T^{2} \) |
| 23 | \( 1 + 48 T + p^{3} T^{2} \) |
| 29 | \( 1 - 34 T + p^{3} T^{2} \) |
| 31 | \( 1 + 320 T + p^{3} T^{2} \) |
| 37 | \( 1 + 310 T + p^{3} T^{2} \) |
| 41 | \( 1 - 10 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 152 T + p^{3} T^{2} \) |
| 47 | \( 1 - 416 T + p^{3} T^{2} \) |
| 53 | \( 1 - 410 T + p^{3} T^{2} \) |
| 59 | \( 1 + 200 T + p^{3} T^{2} \) |
| 61 | \( 1 + 30 T + p^{3} T^{2} \) |
| 67 | \( 1 - 776 T + p^{3} T^{2} \) |
| 71 | \( 1 + 400 T + p^{3} T^{2} \) |
| 73 | \( 1 + 630 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1120 T + p^{3} T^{2} \) |
| 83 | \( 1 - 552 T + p^{3} T^{2} \) |
| 89 | \( 1 + 326 T + p^{3} T^{2} \) |
| 97 | \( 1 + 110 T + p^{3} T^{2} \) |
| 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
−14.14982102184511491235495690038, −13.48964069393090235910718008160, −12.77145120958339874947674081894, −10.65656276211827694304734146052, −9.565636560284052792246753052122, −8.746342392276130690406789274314, −7.43264095817196803524301669000, −5.81522369383697798419272989416, −3.58023957328664455076465662554, −2.24475191344330733293487580316,
2.24475191344330733293487580316, 3.58023957328664455076465662554, 5.81522369383697798419272989416, 7.43264095817196803524301669000, 8.746342392276130690406789274314, 9.565636560284052792246753052122, 10.65656276211827694304734146052, 12.77145120958339874947674081894, 13.48964069393090235910718008160, 14.14982102184511491235495690038
