| L(s) = 1 | − 2-s + 4-s − 3·5-s − 3·7-s − 8-s + 5·9-s + 3·10-s − 3·11-s − 3·13-s + 3·14-s + 16-s − 5·18-s − 3·20-s + 3·22-s + 4·25-s + 3·26-s − 3·28-s + 9·31-s − 32-s + 9·35-s + 5·36-s + 3·40-s − 3·44-s − 15·45-s + 2·49-s − 4·50-s − 3·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s − 1.13·7-s − 0.353·8-s + 5/3·9-s + 0.948·10-s − 0.904·11-s − 0.832·13-s + 0.801·14-s + 1/4·16-s − 1.17·18-s − 0.670·20-s + 0.639·22-s + 4/5·25-s + 0.588·26-s − 0.566·28-s + 1.61·31-s − 0.176·32-s + 1.52·35-s + 5/6·36-s + 0.474·40-s − 0.452·44-s − 2.23·45-s + 2/7·49-s − 0.565·50-s − 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5702787278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5702787278\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 113 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
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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
−9.358505589843119874810751371908, −8.795761381831997565723599828185, −8.143112495800780115404053785138, −7.74664009981249185747033417086, −7.42592214776103680243929691088, −7.00961700365096510426699684874, −6.52044613656007230544660445102, −5.98230758728887215303876950434, −5.09530026968875053920282574615, −4.43959645787744469526960638716, −4.18892115700247061378937312403, −3.19528649018657380927579271669, −2.90921960358122062170412082363, −1.81480663749713697641063333693, −0.55816626785072550843606947503,
0.55816626785072550843606947503, 1.81480663749713697641063333693, 2.90921960358122062170412082363, 3.19528649018657380927579271669, 4.18892115700247061378937312403, 4.43959645787744469526960638716, 5.09530026968875053920282574615, 5.98230758728887215303876950434, 6.52044613656007230544660445102, 7.00961700365096510426699684874, 7.42592214776103680243929691088, 7.74664009981249185747033417086, 8.143112495800780115404053785138, 8.795761381831997565723599828185, 9.358505589843119874810751371908
