| L(s) = 1 | + 2-s − 3-s − 5-s − 6-s − 7-s − 8-s − 10-s + 11-s + 4·13-s − 14-s + 15-s − 16-s − 17-s + 21-s + 22-s − 23-s + 24-s + 25-s + 4·26-s + 27-s + 30-s − 33-s − 34-s + 35-s − 4·39-s + 40-s + 2·41-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s − 5-s − 6-s − 7-s − 8-s − 10-s + 11-s + 4·13-s − 14-s + 15-s − 16-s − 17-s + 21-s + 22-s − 23-s + 24-s + 25-s + 4·26-s + 27-s + 30-s − 33-s − 34-s + 35-s − 4·39-s + 40-s + 2·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3415104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3415104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9534014111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9534014111\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| 11 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_1$ | \( ( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
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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
−9.385208742128943173637913431008, −9.192234853999757280873967759365, −8.795342370125987103020938255833, −8.522782235896556011197429978029, −8.232766441531798270998641283307, −7.68136002484386234198926328245, −6.75257547676380856745129457681, −6.73548163296035874185680777821, −6.24037956665149484637319793692, −6.17440293735885385639400695467, −5.56615818753605502994167801531, −5.47837207377449359768654856137, −4.38065035256337722272442509100, −4.26766811836582954211623897232, −3.97131495345697448457953133767, −3.38854489848684624181388497648, −3.31998325547647789621421358387, −2.46242342553505000627755191418, −1.37179110390895960756657395986, −0.74730451175395455718213868262,
0.74730451175395455718213868262, 1.37179110390895960756657395986, 2.46242342553505000627755191418, 3.31998325547647789621421358387, 3.38854489848684624181388497648, 3.97131495345697448457953133767, 4.26766811836582954211623897232, 4.38065035256337722272442509100, 5.47837207377449359768654856137, 5.56615818753605502994167801531, 6.17440293735885385639400695467, 6.24037956665149484637319793692, 6.73548163296035874185680777821, 6.75257547676380856745129457681, 7.68136002484386234198926328245, 8.232766441531798270998641283307, 8.522782235896556011197429978029, 8.795342370125987103020938255833, 9.192234853999757280873967759365, 9.385208742128943173637913431008
