| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 2·9-s − 14-s + 16-s + 10·17-s − 2·18-s + 4·23-s − 2·25-s − 28-s + 8·31-s + 32-s + 10·34-s − 2·36-s + 2·41-s + 4·46-s − 8·47-s + 49-s − 2·50-s − 56-s + 8·62-s + 2·63-s + 64-s + 10·68-s − 20·71-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.267·14-s + 1/4·16-s + 2.42·17-s − 0.471·18-s + 0.834·23-s − 2/5·25-s − 0.188·28-s + 1.43·31-s + 0.176·32-s + 1.71·34-s − 1/3·36-s + 0.312·41-s + 0.589·46-s − 1.16·47-s + 1/7·49-s − 0.282·50-s − 0.133·56-s + 1.01·62-s + 0.251·63-s + 1/8·64-s + 1.21·68-s − 2.37·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.008628703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.008628703\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 16 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−10.26198625768805273878719158597, −9.807283911451784848867871077047, −9.233870777277990810646909989035, −8.626454361062853678938537989862, −7.87947203505069177916060814553, −7.72017961903121285224118371713, −6.93295761301336712015981974235, −6.28456104405352899088791784904, −5.84236365237513916779554422457, −5.27743924328440857639324718167, −4.72166922153586628950292773141, −3.80670019043265087333146656334, −3.17442428449498352846146247269, −2.69616292609918723606429253208, −1.27833369925900360338012527167,
1.27833369925900360338012527167, 2.69616292609918723606429253208, 3.17442428449498352846146247269, 3.80670019043265087333146656334, 4.72166922153586628950292773141, 5.27743924328440857639324718167, 5.84236365237513916779554422457, 6.28456104405352899088791784904, 6.93295761301336712015981974235, 7.72017961903121285224118371713, 7.87947203505069177916060814553, 8.626454361062853678938537989862, 9.233870777277990810646909989035, 9.807283911451784848867871077047, 10.26198625768805273878719158597
