L(s) = 1 | + 2·3-s + 9-s − 4·19-s − 10·25-s − 4·27-s + 12·29-s − 16·43-s − 24·47-s + 49-s − 12·53-s − 8·57-s + 8·67-s + 4·73-s − 20·75-s − 11·81-s + 24·87-s − 20·97-s − 22·121-s + 127-s − 32·129-s + 131-s + 137-s + 139-s − 48·141-s + 2·147-s + 149-s + 151-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1/3·9-s − 0.917·19-s − 2·25-s − 0.769·27-s + 2.22·29-s − 2.43·43-s − 3.50·47-s + 1/7·49-s − 1.64·53-s − 1.05·57-s + 0.977·67-s + 0.468·73-s − 2.30·75-s − 1.22·81-s + 2.57·87-s − 2.03·97-s − 2·121-s + 0.0887·127-s − 2.81·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4.04·141-s + 0.164·147-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.222985047096640302551257734717, −7.967527986115067076241988284772, −7.87021772021814993834479658639, −6.84458804347565715458286520604, −6.47781377385923123584313505983, −6.34752255234086455771568318662, −5.36663802373794330843690451326, −5.05202336643031434355439952711, −4.32309265725798888275246837851, −3.93522119575008455702519349301, −3.07302685938302839530976846531, −3.04185374086143547449059084416, −1.98782431119292345428047370492, −1.64950502134773404742708641242, 0,
1.64950502134773404742708641242, 1.98782431119292345428047370492, 3.04185374086143547449059084416, 3.07302685938302839530976846531, 3.93522119575008455702519349301, 4.32309265725798888275246837851, 5.05202336643031434355439952711, 5.36663802373794330843690451326, 6.34752255234086455771568318662, 6.47781377385923123584313505983, 6.84458804347565715458286520604, 7.87021772021814993834479658639, 7.967527986115067076241988284772, 8.222985047096640302551257734717