| L(s) = 1 | + 4·7-s + 4·11-s − 13-s + 17-s − 4·19-s − 8·23-s + 2·29-s − 4·31-s + 6·37-s + 6·41-s + 4·43-s − 4·47-s + 9·49-s + 14·53-s − 10·61-s + 12·67-s + 8·71-s + 10·73-s + 16·77-s + 8·79-s − 8·83-s − 6·89-s − 4·91-s + 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 1.20·11-s − 0.277·13-s + 0.242·17-s − 0.917·19-s − 1.66·23-s + 0.371·29-s − 0.718·31-s + 0.986·37-s + 0.937·41-s + 0.609·43-s − 0.583·47-s + 9/7·49-s + 1.92·53-s − 1.28·61-s + 1.46·67-s + 0.949·71-s + 1.17·73-s + 1.82·77-s + 0.900·79-s − 0.878·83-s − 0.635·89-s − 0.419·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.617081404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.617081404\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| 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
−12.37556294404798, −11.93956788107896, −11.50623397219394, −11.21415536785114, −10.65251828570347, −10.31619259182902, −9.627788477992536, −9.310609228704907, −8.764166304467003, −8.310921562800337, −7.834629048246412, −7.644852465790913, −6.890612259191068, −6.413955986976003, −6.029365275540150, −5.369361351644641, −5.031094448309559, −4.285953670382680, −4.025028788084217, −3.723641125047582, −2.596662226626724, −2.298298224496133, −1.668736621678799, −1.202348687526950, −0.4981586817488962,
0.4981586817488962, 1.202348687526950, 1.668736621678799, 2.298298224496133, 2.596662226626724, 3.723641125047582, 4.025028788084217, 4.285953670382680, 5.031094448309559, 5.369361351644641, 6.029365275540150, 6.413955986976003, 6.890612259191068, 7.644852465790913, 7.834629048246412, 8.310921562800337, 8.764166304467003, 9.310609228704907, 9.627788477992536, 10.31619259182902, 10.65251828570347, 11.21415536785114, 11.50623397219394, 11.93956788107896, 12.37556294404798
