| L(s) = 1 | − 4-s − 2·7-s − 3·9-s + 16-s − 10·25-s + 2·28-s + 3·36-s − 4·37-s + 20·43-s − 3·49-s + 6·63-s − 64-s + 16·67-s + 4·79-s + 9·81-s + 10·100-s − 4·109-s − 2·112-s − 121-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 4·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.755·7-s − 9-s + 1/4·16-s − 2·25-s + 0.377·28-s + 1/2·36-s − 0.657·37-s + 3.04·43-s − 3/7·49-s + 0.755·63-s − 1/8·64-s + 1.95·67-s + 0.450·79-s + 81-s + 100-s − 0.383·109-s − 0.188·112-s − 0.0909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/4·144-s + 0.328·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8634849719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8634849719\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.125209855375335254699211520970, −8.611900847015872064943536180579, −8.117534024186648310878629031896, −7.70655723529328114668098956620, −7.22979759196250675579890868762, −6.47219176192832992122576869145, −6.14972858975095139308559764943, −5.53709302618949700664822828730, −5.31886809335651649946095510331, −4.37463318132185984279194975708, −3.92749254031617109541017214946, −3.36990487566990159455496947455, −2.67581994958300247566042268968, −1.97811754946896275223649615219, −0.56780151834311949444328435918,
0.56780151834311949444328435918, 1.97811754946896275223649615219, 2.67581994958300247566042268968, 3.36990487566990159455496947455, 3.92749254031617109541017214946, 4.37463318132185984279194975708, 5.31886809335651649946095510331, 5.53709302618949700664822828730, 6.14972858975095139308559764943, 6.47219176192832992122576869145, 7.22979759196250675579890868762, 7.70655723529328114668098956620, 8.117534024186648310878629031896, 8.611900847015872064943536180579, 9.125209855375335254699211520970
