| L(s) = 1 | + 4-s + 7-s + 16-s + 6·17-s − 10·25-s + 28-s + 4·37-s − 12·41-s − 2·43-s − 12·47-s + 49-s − 6·59-s + 64-s − 26·67-s + 6·68-s − 20·79-s − 24·83-s + 30·89-s − 10·100-s + 36·101-s − 32·109-s + 112-s + 6·119-s − 22·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.377·7-s + 1/4·16-s + 1.45·17-s − 2·25-s + 0.188·28-s + 0.657·37-s − 1.87·41-s − 0.304·43-s − 1.75·47-s + 1/7·49-s − 0.781·59-s + 1/8·64-s − 3.17·67-s + 0.727·68-s − 2.25·79-s − 2.63·83-s + 3.17·89-s − 100-s + 3.58·101-s − 3.06·109-s + 0.0944·112-s + 0.550·119-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 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 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.66081859530557802153309309893, −7.62887873867370361852526319092, −7.21285497786280286801180495079, −6.33738862277121156447234649392, −6.29670817383847541122613475379, −5.61879984028395623126782300752, −5.36650684226892599309211492342, −4.64577805751899849665319383092, −4.30645127647783484429164317499, −3.38877464404185294878323159152, −3.34773629764867655679280376645, −2.54401642319011399557920759889, −1.68869994047015348511995782895, −1.43423414088871078406280124121, 0,
1.43423414088871078406280124121, 1.68869994047015348511995782895, 2.54401642319011399557920759889, 3.34773629764867655679280376645, 3.38877464404185294878323159152, 4.30645127647783484429164317499, 4.64577805751899849665319383092, 5.36650684226892599309211492342, 5.61879984028395623126782300752, 6.29670817383847541122613475379, 6.33738862277121156447234649392, 7.21285497786280286801180495079, 7.62887873867370361852526319092, 7.66081859530557802153309309893
