| L(s) = 1 | − 2·4-s − 12·13-s + 4·16-s + 24·19-s + 32·25-s + 56·31-s − 4·37-s − 120·43-s − 98·49-s + 24·52-s − 48·61-s − 8·64-s − 160·67-s + 204·73-s − 48·76-s − 240·79-s + 64·97-s − 64·100-s − 56·103-s − 84·109-s − 112·124-s + 127-s + 131-s + 137-s + 139-s + 8·148-s + 149-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.923·13-s + 1/4·16-s + 1.26·19-s + 1.27·25-s + 1.80·31-s − 0.108·37-s − 2.79·43-s − 2·49-s + 6/13·52-s − 0.786·61-s − 1/8·64-s − 2.38·67-s + 2.79·73-s − 0.631·76-s − 3.03·79-s + 0.659·97-s − 0.639·100-s − 0.543·103-s − 0.770·109-s − 0.903·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 2/37·148-s + 0.00671·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4743684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4743684 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7922603504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7922603504\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 32 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 560 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 12 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 770 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 496 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 28 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2480 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 60 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3266 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4736 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4370 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 80 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 102 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 120 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 158 T + p^{2} T^{2} )( 1 + 158 T + p^{2} T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 6208 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{2} \) |
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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
−9.275291318356636801710762466684, −8.469658600833408357812293756765, −8.439432673383146404517714552109, −7.969170876125092629155926964637, −7.62581175633449385619332284287, −7.00630768452701734143043439560, −6.86846516334192370831511171547, −6.40254780919141205896275091584, −5.91049386813209562201852534295, −5.43825066581552374848075182436, −4.89206422396007923807569879850, −4.73934294781835678877885185086, −4.50433912531857859305848509611, −3.55998820675644067137365687511, −3.32379564512647523434736869776, −2.84630832809434632701961203915, −2.39270603868505092326614327401, −1.41267963165745768777209820675, −1.25097456145795971783794010288, −0.23474500242067484982669812839,
0.23474500242067484982669812839, 1.25097456145795971783794010288, 1.41267963165745768777209820675, 2.39270603868505092326614327401, 2.84630832809434632701961203915, 3.32379564512647523434736869776, 3.55998820675644067137365687511, 4.50433912531857859305848509611, 4.73934294781835678877885185086, 4.89206422396007923807569879850, 5.43825066581552374848075182436, 5.91049386813209562201852534295, 6.40254780919141205896275091584, 6.86846516334192370831511171547, 7.00630768452701734143043439560, 7.62581175633449385619332284287, 7.969170876125092629155926964637, 8.439432673383146404517714552109, 8.469658600833408357812293756765, 9.275291318356636801710762466684
