| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s − 2·5-s + 4·6-s + 4·8-s + 3·9-s − 4·10-s − 2·11-s + 6·12-s − 6·13-s − 4·15-s + 5·16-s − 2·17-s + 6·18-s − 2·19-s − 6·20-s − 4·22-s − 10·23-s + 8·24-s + 3·25-s − 12·26-s + 4·27-s − 10·29-s − 8·30-s − 6·31-s + 6·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s + 1.41·8-s + 9-s − 1.26·10-s − 0.603·11-s + 1.73·12-s − 1.66·13-s − 1.03·15-s + 5/4·16-s − 0.485·17-s + 1.41·18-s − 0.458·19-s − 1.34·20-s − 0.852·22-s − 2.08·23-s + 1.63·24-s + 3/5·25-s − 2.35·26-s + 0.769·27-s − 1.85·29-s − 1.46·30-s − 1.07·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 10 T + 54 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 54 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 66 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 86 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 22 T + 238 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 142 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 226 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 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
−7.59565933840764912804023777553, −7.52758535829043867382635461418, −7.27604784579785602623909848837, −7.26897996237402212074464893435, −6.43963903677969449862242740243, −6.14956528912217179628436673726, −5.75281948653059634820000118982, −5.32907371979394353898393743050, −4.87569358316474075803455692966, −4.63048934412518958920120916198, −4.10662162864964658369848263162, −4.00178855072732578898442483202, −3.48420996313208852283539560775, −3.18949179672656999399838934485, −2.61070292968244389297295922065, −2.38079235788997036913468508395, −1.75207412152694343645837694563, −1.66442771112067544496094532083, 0, 0,
1.66442771112067544496094532083, 1.75207412152694343645837694563, 2.38079235788997036913468508395, 2.61070292968244389297295922065, 3.18949179672656999399838934485, 3.48420996313208852283539560775, 4.00178855072732578898442483202, 4.10662162864964658369848263162, 4.63048934412518958920120916198, 4.87569358316474075803455692966, 5.32907371979394353898393743050, 5.75281948653059634820000118982, 6.14956528912217179628436673726, 6.43963903677969449862242740243, 7.26897996237402212074464893435, 7.27604784579785602623909848837, 7.52758535829043867382635461418, 7.59565933840764912804023777553
