| L(s) = 1 | + 2-s − 4-s − 3·8-s + 9-s − 16-s − 12·17-s + 18-s + 8·23-s − 6·25-s + 16·31-s + 5·32-s − 12·34-s − 36-s − 4·41-s + 8·46-s + 24·47-s − 14·49-s − 6·50-s + 16·62-s + 7·64-s + 12·68-s − 3·72-s + 20·73-s + 81-s − 4·82-s − 4·89-s − 8·92-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s + 1/3·9-s − 1/4·16-s − 2.91·17-s + 0.235·18-s + 1.66·23-s − 6/5·25-s + 2.87·31-s + 0.883·32-s − 2.05·34-s − 1/6·36-s − 0.624·41-s + 1.17·46-s + 3.50·47-s − 2·49-s − 0.848·50-s + 2.03·62-s + 7/8·64-s + 1.45·68-s − 0.353·72-s + 2.34·73-s + 1/9·81-s − 0.441·82-s − 0.423·89-s − 0.834·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207936 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207936 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.665775316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.665775316\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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
−9.032116632714081125535639739797, −8.656147919084617168059664238609, −8.263513432862467471567669978548, −7.62692916507233066692617069633, −6.96313967889007715870809714116, −6.46787589567348650226556085189, −6.30766736214003560110551425788, −5.52157301147752958888078955745, −4.83501817913468223446739446530, −4.56991321080597356703354311172, −4.16209472915564587197687363565, −3.44365518362789223021999993571, −2.66586826806470965226318272179, −2.15027495389545596722403606493, −0.72964391205275968179204890320,
0.72964391205275968179204890320, 2.15027495389545596722403606493, 2.66586826806470965226318272179, 3.44365518362789223021999993571, 4.16209472915564587197687363565, 4.56991321080597356703354311172, 4.83501817913468223446739446530, 5.52157301147752958888078955745, 6.30766736214003560110551425788, 6.46787589567348650226556085189, 6.96313967889007715870809714116, 7.62692916507233066692617069633, 8.263513432862467471567669978548, 8.656147919084617168059664238609, 9.032116632714081125535639739797
