| L(s) = 1 | + 3-s − 2·9-s + 4·13-s − 2·19-s − 25-s − 5·27-s − 14·31-s − 2·37-s + 4·39-s − 8·43-s − 2·57-s − 2·61-s − 14·67-s − 2·73-s − 75-s − 26·79-s + 81-s − 14·93-s − 20·97-s + 22·103-s − 2·109-s − 2·111-s − 8·117-s − 13·121-s + 127-s − 8·129-s + 131-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s + 1.10·13-s − 0.458·19-s − 1/5·25-s − 0.962·27-s − 2.51·31-s − 0.328·37-s + 0.640·39-s − 1.21·43-s − 0.264·57-s − 0.256·61-s − 1.71·67-s − 0.234·73-s − 0.115·75-s − 2.92·79-s + 1/9·81-s − 1.45·93-s − 2.03·97-s + 2.16·103-s − 0.191·109-s − 0.189·111-s − 0.739·117-s − 1.18·121-s + 0.0887·127-s − 0.704·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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
−8.646784595072977282306137209646, −8.262637018939911096603858847029, −7.51155146227211557820915407148, −7.27657735328498763814943158038, −6.67413798525527807153917855846, −6.02455306491682974275355684196, −5.71711213824960626307579454428, −5.28434328079604079625535316421, −4.44529829557613531132250015524, −3.98660375345473223094112675622, −3.31407617989174490462667281240, −3.04451713874417289677681835886, −2.04448974040096779312937543400, −1.54505867526772913791014367907, 0,
1.54505867526772913791014367907, 2.04448974040096779312937543400, 3.04451713874417289677681835886, 3.31407617989174490462667281240, 3.98660375345473223094112675622, 4.44529829557613531132250015524, 5.28434328079604079625535316421, 5.71711213824960626307579454428, 6.02455306491682974275355684196, 6.67413798525527807153917855846, 7.27657735328498763814943158038, 7.51155146227211557820915407148, 8.262637018939911096603858847029, 8.646784595072977282306137209646
