| L(s) = 1 | − 2·2-s + 8·3-s + 4·4-s − 14·5-s − 16·6-s − 7·7-s − 8·8-s + 37·9-s + 28·10-s − 28·11-s + 32·12-s + 18·13-s + 14·14-s − 112·15-s + 16·16-s + 74·17-s − 74·18-s + 80·19-s − 56·20-s − 56·21-s + 56·22-s − 112·23-s − 64·24-s + 71·25-s − 36·26-s + 80·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.53·3-s + 1/2·4-s − 1.25·5-s − 1.08·6-s − 0.377·7-s − 0.353·8-s + 1.37·9-s + 0.885·10-s − 0.767·11-s + 0.769·12-s + 0.384·13-s + 0.267·14-s − 1.92·15-s + 1/4·16-s + 1.05·17-s − 0.968·18-s + 0.965·19-s − 0.626·20-s − 0.581·21-s + 0.542·22-s − 1.01·23-s − 0.544·24-s + 0.567·25-s − 0.271·26-s + 0.570·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9193067426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9193067426\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( 1 - 8 T + p^{3} T^{2} \) |
| 5 | \( 1 + 14 T + p^{3} T^{2} \) |
| 11 | \( 1 + 28 T + p^{3} T^{2} \) |
| 13 | \( 1 - 18 T + p^{3} T^{2} \) |
| 17 | \( 1 - 74 T + p^{3} T^{2} \) |
| 19 | \( 1 - 80 T + p^{3} T^{2} \) |
| 23 | \( 1 + 112 T + p^{3} T^{2} \) |
| 29 | \( 1 - 190 T + p^{3} T^{2} \) |
| 31 | \( 1 - 72 T + p^{3} T^{2} \) |
| 37 | \( 1 + 346 T + p^{3} T^{2} \) |
| 41 | \( 1 - 162 T + p^{3} T^{2} \) |
| 43 | \( 1 + 412 T + p^{3} T^{2} \) |
| 47 | \( 1 - 24 T + p^{3} T^{2} \) |
| 53 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 59 | \( 1 + 200 T + p^{3} T^{2} \) |
| 61 | \( 1 + 198 T + p^{3} T^{2} \) |
| 67 | \( 1 + 716 T + p^{3} T^{2} \) |
| 71 | \( 1 - 392 T + p^{3} T^{2} \) |
| 73 | \( 1 - 538 T + p^{3} T^{2} \) |
| 79 | \( 1 - 240 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1072 T + p^{3} T^{2} \) |
| 89 | \( 1 - 810 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1354 T + p^{3} T^{2} \) |
| 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
−19.35333994878018611468238019662, −18.37630389516892545263242891991, −16.11376996869225351950573125836, −15.36902088010492813544845878005, −13.88019434035864167717373197052, −12.07836169532718105087929262676, −10.06164256849421362540360180538, −8.448050551275855537988611024596, −7.57224949670005631561767955805, −3.31980664588488811136973402542,
3.31980664588488811136973402542, 7.57224949670005631561767955805, 8.448050551275855537988611024596, 10.06164256849421362540360180538, 12.07836169532718105087929262676, 13.88019434035864167717373197052, 15.36902088010492813544845878005, 16.11376996869225351950573125836, 18.37630389516892545263242891991, 19.35333994878018611468238019662
