| L(s) = 1 | − 4·4-s − 12·11-s + 16·16-s − 88·19-s + 252·29-s − 488·31-s + 960·41-s + 48·44-s + 490·49-s + 1.06e3·59-s + 724·61-s − 64·64-s + 1.94e3·71-s + 352·76-s − 2.48e3·79-s − 1.94e3·89-s + 3.01e3·101-s − 1.39e3·109-s − 1.00e3·116-s − 2.55e3·121-s + 1.95e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.328·11-s + 1/4·16-s − 1.06·19-s + 1.61·29-s − 2.82·31-s + 3.65·41-s + 0.164·44-s + 10/7·49-s + 2.35·59-s + 1.51·61-s − 1/8·64-s + 3.24·71-s + 0.531·76-s − 3.54·79-s − 2.31·89-s + 2.96·101-s − 1.22·109-s − 0.806·116-s − 1.91·121-s + 1.41·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.698128260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.698128260\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 10 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 230 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3742 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 44 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 9934 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 126 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 244 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 8890 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 480 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 148198 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 152354 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 231190 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 534 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 362 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 p^{2} T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 972 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 557134 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1244 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 986758 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 972 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1823230 T^{2} + p^{6} T^{4} \) |
| 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
−10.79055908212195378412944646315, −10.50660331137782613243107151089, −10.02169034745124019765302137638, −9.515525246392198693287779669934, −9.042088177048726839310535444920, −8.727113518480559458849071489633, −8.199210416913976268165464284598, −7.79970490284372622912019963914, −7.02497890527338251595744714774, −6.99354451089136047503674363806, −6.06993041983229543660622433247, −5.64816651530711963779472048450, −5.28035131384639514186325059989, −4.53450362904211865588275318406, −3.95883014500040833368426778375, −3.74001390302376578162910707036, −2.45923174085190700919372007706, −2.44919589353584210765304264684, −1.19952123479345551092621614185, −0.46752539479720959482519751673,
0.46752539479720959482519751673, 1.19952123479345551092621614185, 2.44919589353584210765304264684, 2.45923174085190700919372007706, 3.74001390302376578162910707036, 3.95883014500040833368426778375, 4.53450362904211865588275318406, 5.28035131384639514186325059989, 5.64816651530711963779472048450, 6.06993041983229543660622433247, 6.99354451089136047503674363806, 7.02497890527338251595744714774, 7.79970490284372622912019963914, 8.199210416913976268165464284598, 8.727113518480559458849071489633, 9.042088177048726839310535444920, 9.515525246392198693287779669934, 10.02169034745124019765302137638, 10.50660331137782613243107151089, 10.79055908212195378412944646315
