L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s − 11-s − 4·13-s − 2·14-s + 16-s − 6·17-s + 4·19-s − 22-s − 5·25-s − 4·26-s − 2·28-s − 6·29-s + 8·31-s + 32-s − 6·34-s + 10·37-s + 4·38-s − 6·41-s − 8·43-s − 44-s + 6·47-s − 3·49-s − 5·50-s − 4·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s − 0.301·11-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s − 0.213·22-s − 25-s − 0.784·26-s − 0.377·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s − 1.02·34-s + 1.64·37-s + 0.648·38-s − 0.937·41-s − 1.21·43-s − 0.150·44-s + 0.875·47-s − 3/7·49-s − 0.707·50-s − 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104742 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104742 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.151502420\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.151502420\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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
−13.71526802454961, −13.20999999333811, −12.87053868000510, −12.28137653618769, −11.82381323810037, −11.32643994624719, −11.01155663469963, −10.03657919598381, −9.908977030169332, −9.477582599962136, −8.715979269464900, −8.189657741190867, −7.453552548139540, −7.229696629896775, −6.531974945554586, −6.127254348178527, −5.532078231013295, −4.969400737871550, −4.394399515538868, −3.989449462765231, −3.113380050395691, −2.771138423366196, −2.159984409268160, −1.421397039625412, −0.2792167317276017,
0.2792167317276017, 1.421397039625412, 2.159984409268160, 2.771138423366196, 3.113380050395691, 3.989449462765231, 4.394399515538868, 4.969400737871550, 5.532078231013295, 6.127254348178527, 6.531974945554586, 7.229696629896775, 7.453552548139540, 8.189657741190867, 8.715979269464900, 9.477582599962136, 9.908977030169332, 10.03657919598381, 11.01155663469963, 11.32643994624719, 11.82381323810037, 12.28137653618769, 12.87053868000510, 13.20999999333811, 13.71526802454961