| L(s) = 1 | + 6·3-s − 16·4-s − 10·5-s + 27·9-s − 96·12-s − 60·15-s + 192·16-s + 160·20-s − 144·23-s + 75·25-s + 108·27-s − 40·31-s − 432·36-s + 428·37-s − 270·45-s − 528·47-s + 1.15e3·48-s + 34·49-s + 156·53-s + 960·59-s + 960·60-s − 2.04e3·64-s − 1.04e3·67-s − 864·69-s + 984·71-s + 450·75-s − 1.92e3·80-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 2·4-s − 0.894·5-s + 9-s − 2.30·12-s − 1.03·15-s + 3·16-s + 1.78·20-s − 1.30·23-s + 3/5·25-s + 0.769·27-s − 0.231·31-s − 2·36-s + 1.90·37-s − 0.894·45-s − 1.63·47-s + 3.46·48-s + 0.0991·49-s + 0.404·53-s + 2.11·59-s + 2.06·60-s − 4·64-s − 1.91·67-s − 1.50·69-s + 1.64·71-s + 0.692·75-s − 2.68·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 34 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 3674 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6946 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 7238 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 72 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 38342 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 214 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 137122 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 100694 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 264 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 78 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 480 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 442442 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 524 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 492 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 103966 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 898958 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 795094 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1206 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 1186 T + p^{3} T^{2} )^{2} \) |
| 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
−8.505339789761480325570388560840, −8.396325060680453253113225068510, −8.058527055750189401894786916448, −7.88609479581138711086273790967, −7.21391746325451279632322316356, −7.04001468293727886004973629698, −6.12364602889260379701964746438, −5.96282739211297405241955261438, −5.13268025368723137465758497849, −5.04072163540495502989633507945, −4.22418549880912832545368315040, −4.20828066892846926766713439507, −3.80469700753090062589295519741, −3.38357280892382716259717909198, −2.80075488410104803892435845838, −2.30949000764317530133897246326, −1.33096072105625854039277987182, −1.03995354163611040087452610159, 0, 0,
1.03995354163611040087452610159, 1.33096072105625854039277987182, 2.30949000764317530133897246326, 2.80075488410104803892435845838, 3.38357280892382716259717909198, 3.80469700753090062589295519741, 4.20828066892846926766713439507, 4.22418549880912832545368315040, 5.04072163540495502989633507945, 5.13268025368723137465758497849, 5.96282739211297405241955261438, 6.12364602889260379701964746438, 7.04001468293727886004973629698, 7.21391746325451279632322316356, 7.88609479581138711086273790967, 8.058527055750189401894786916448, 8.396325060680453253113225068510, 8.505339789761480325570388560840
