| L(s) = 1 | − 2·2-s + 3·4-s + 7-s − 4·8-s − 2·14-s + 5·16-s + 18·23-s − 10·25-s + 3·28-s + 6·29-s − 6·32-s + 4·37-s − 2·43-s − 36·46-s + 49-s + 20·50-s − 6·53-s − 4·56-s − 12·58-s + 7·64-s − 26·67-s − 18·71-s − 8·74-s − 20·79-s + 4·86-s + 54·92-s − 2·98-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.377·7-s − 1.41·8-s − 0.534·14-s + 5/4·16-s + 3.75·23-s − 2·25-s + 0.566·28-s + 1.11·29-s − 1.06·32-s + 0.657·37-s − 0.304·43-s − 5.30·46-s + 1/7·49-s + 2.82·50-s − 0.824·53-s − 0.534·56-s − 1.57·58-s + 7/8·64-s − 3.17·67-s − 2.13·71-s − 0.929·74-s − 2.25·79-s + 0.431·86-s + 5.62·92-s − 0.202·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000188 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000188 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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
−7.77295226765590417072124632150, −7.62887873867370361852526319092, −7.21285497786280286801180495079, −6.72994634489814968752118634163, −6.33738862277121156447234649392, −5.74408802320171077841105153966, −5.36650684226892599309211492342, −4.64577805751899849665319383092, −4.35441080626390222166206464206, −3.38877464404185294878323159152, −2.89963569250000254763681159752, −2.54401642319011399557920759889, −1.43423414088871078406280124121, −1.25841763926025585348683837537, 0,
1.25841763926025585348683837537, 1.43423414088871078406280124121, 2.54401642319011399557920759889, 2.89963569250000254763681159752, 3.38877464404185294878323159152, 4.35441080626390222166206464206, 4.64577805751899849665319383092, 5.36650684226892599309211492342, 5.74408802320171077841105153966, 6.33738862277121156447234649392, 6.72994634489814968752118634163, 7.21285497786280286801180495079, 7.62887873867370361852526319092, 7.77295226765590417072124632150
