| L(s) = 1 | − 178·13-s + 250·25-s − 866·37-s − 397·49-s − 1.80e3·61-s + 542·73-s − 3.70e3·97-s + 1.29e3·109-s − 2.66e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.93e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 3.79·13-s + 2·25-s − 3.84·37-s − 1.15·49-s − 3.78·61-s + 0.868·73-s − 3.87·97-s + 1.13·109-s − 2·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 8.81·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 17 T + p^{3} T^{2} )( 1 + 17 T + p^{3} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 89 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 107 T + p^{3} T^{2} )( 1 + 107 T + p^{3} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 308 T + p^{3} T^{2} )( 1 + 308 T + p^{3} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 433 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 520 T + p^{3} T^{2} )( 1 + 520 T + p^{3} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 901 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 1007 T + p^{3} T^{2} )( 1 + 1007 T + p^{3} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 271 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 503 T + p^{3} T^{2} )( 1 + 503 T + p^{3} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 1853 T + p^{3} T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.28657176388879190146371166068, −10.27664903676889688649347679949, −9.560945575164827817792960325744, −9.234110190151309942324843667191, −8.817232735287914562414151096961, −8.176653614018976379689498973269, −7.55239573072325795768770326307, −7.34297777075260599005533846303, −6.72967014436714130385523910118, −6.60910255470084192685458328411, −5.37670338493954760822069076724, −5.19403733282421811535969846355, −4.79149732185335811587906606616, −4.29648735786399857592366647550, −3.17222073629957670720887843289, −2.92025749855875316428798863863, −2.17654932031609721244871505393, −1.48692194546584901920359089368, 0, 0,
1.48692194546584901920359089368, 2.17654932031609721244871505393, 2.92025749855875316428798863863, 3.17222073629957670720887843289, 4.29648735786399857592366647550, 4.79149732185335811587906606616, 5.19403733282421811535969846355, 5.37670338493954760822069076724, 6.60910255470084192685458328411, 6.72967014436714130385523910118, 7.34297777075260599005533846303, 7.55239573072325795768770326307, 8.176653614018976379689498973269, 8.817232735287914562414151096961, 9.234110190151309942324843667191, 9.560945575164827817792960325744, 10.27664903676889688649347679949, 10.28657176388879190146371166068
