L(s) = 1 | + 3·4-s + 6·13-s + 5·16-s − 8·17-s − 16·23-s − 25-s + 16·29-s + 16·43-s − 2·49-s + 18·52-s + 24·53-s − 4·61-s + 3·64-s − 24·68-s − 16·79-s − 48·92-s − 3·100-s + 32·101-s + 8·103-s + 16·107-s + 8·113-s + 48·116-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | + 3/2·4-s + 1.66·13-s + 5/4·16-s − 1.94·17-s − 3.33·23-s − 1/5·25-s + 2.97·29-s + 2.43·43-s − 2/7·49-s + 2.49·52-s + 3.29·53-s − 0.512·61-s + 3/8·64-s − 2.91·68-s − 1.80·79-s − 5.00·92-s − 0.299·100-s + 3.18·101-s + 0.788·103-s + 1.54·107-s + 0.752·113-s + 4.45·116-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.743332040\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.743332040\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 178 T^{2} + p^{2} T^{4} \) |
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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
−11.07349432980446025995960795723, −10.38559214178587156418313538205, −10.20997781130083069713440051899, −9.953440349825541674078577629513, −8.859929457988981522701384298496, −8.696605152572893461376411603374, −8.424628054204083693014282226639, −7.70627867723409188554045580641, −7.35957728137547715617732859206, −6.80451348856461094121887567165, −6.24939274736086820220532197566, −5.97541394162336676655580748258, −5.94857462281854737758207200707, −4.72929811400158416163632280812, −4.19401162936134591544195603344, −3.86655853655184341389840404198, −3.00402961081401192024180662387, −2.17986464670808874121040457053, −2.14955067662077876701430927224, −0.950734341939368621729886097054,
0.950734341939368621729886097054, 2.14955067662077876701430927224, 2.17986464670808874121040457053, 3.00402961081401192024180662387, 3.86655853655184341389840404198, 4.19401162936134591544195603344, 4.72929811400158416163632280812, 5.94857462281854737758207200707, 5.97541394162336676655580748258, 6.24939274736086820220532197566, 6.80451348856461094121887567165, 7.35957728137547715617732859206, 7.70627867723409188554045580641, 8.424628054204083693014282226639, 8.696605152572893461376411603374, 8.859929457988981522701384298496, 9.953440349825541674078577629513, 10.20997781130083069713440051899, 10.38559214178587156418313538205, 11.07349432980446025995960795723