| L(s) = 1 | + 2-s + 4-s + 8-s + 9-s + 16-s + 2·17-s + 18-s − 6·25-s + 32-s + 2·34-s + 36-s + 16·47-s + 10·49-s − 6·50-s + 64-s + 2·68-s + 72-s + 81-s + 20·89-s + 16·94-s + 10·98-s − 6·100-s + 32·103-s − 22·121-s + 127-s + 128-s + 131-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1/3·9-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 6/5·25-s + 0.176·32-s + 0.342·34-s + 1/6·36-s + 2.33·47-s + 10/7·49-s − 0.848·50-s + 1/8·64-s + 0.242·68-s + 0.117·72-s + 1/9·81-s + 2.11·89-s + 1.65·94-s + 1.01·98-s − 3/5·100-s + 3.15·103-s − 2·121-s + 0.0887·127-s + 0.0883·128-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.990522735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.990522735\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−8.839195003001577747571922628762, −8.199397133263483419221021888103, −7.59266086318036482333944944175, −7.47836930577860038386915655621, −6.90369365410745389037013744241, −6.25353947028937099177490903072, −5.90482930646410958798074482335, −5.42848051451252561002388649254, −4.90421879599021250675981877716, −4.23628918591762555005026706755, −3.87002081775942335033180550859, −3.30466060201462864038082665355, −2.49022410853351334524631782583, −1.97065925433832048403434650419, −0.918455563284789218005293309445,
0.918455563284789218005293309445, 1.97065925433832048403434650419, 2.49022410853351334524631782583, 3.30466060201462864038082665355, 3.87002081775942335033180550859, 4.23628918591762555005026706755, 4.90421879599021250675981877716, 5.42848051451252561002388649254, 5.90482930646410958798074482335, 6.25353947028937099177490903072, 6.90369365410745389037013744241, 7.47836930577860038386915655621, 7.59266086318036482333944944175, 8.199397133263483419221021888103, 8.839195003001577747571922628762
