| L(s) = 1 | + 2-s − 3-s + 4-s + 3·5-s − 6-s + 8-s + 9-s + 3·10-s − 12-s − 3·15-s + 16-s + 18-s + 3·20-s − 12·23-s − 24-s + 2·25-s − 27-s − 8·29-s − 3·30-s + 32-s + 36-s + 3·40-s + 12·43-s + 3·45-s − 12·46-s − 48-s + 6·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.288·12-s − 0.774·15-s + 1/4·16-s + 0.235·18-s + 0.670·20-s − 2.50·23-s − 0.204·24-s + 2/5·25-s − 0.192·27-s − 1.48·29-s − 0.547·30-s + 0.176·32-s + 1/6·36-s + 0.474·40-s + 1.82·43-s + 0.447·45-s − 1.76·46-s − 0.144·48-s + 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17280 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17280 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.684534961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.684534961\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| good | 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 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 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−10.99351839993430246414427951188, −10.37772506144100212875288646617, −10.15914639422395200692225500995, −9.466310869162471744831857869431, −9.043180052806949082272619396875, −8.102222126098087303874229299528, −7.51159296698756849818980636470, −6.88600423079228429221319618667, −6.04713192253452202793297618763, −5.78856133280022125700095916440, −5.43432594163021746011280461785, −4.31761926207930601749254749266, −3.86444018024763414741997576626, −2.50583976800103525719285104860, −1.79002829831261169734649175248,
1.79002829831261169734649175248, 2.50583976800103525719285104860, 3.86444018024763414741997576626, 4.31761926207930601749254749266, 5.43432594163021746011280461785, 5.78856133280022125700095916440, 6.04713192253452202793297618763, 6.88600423079228429221319618667, 7.51159296698756849818980636470, 8.102222126098087303874229299528, 9.043180052806949082272619396875, 9.466310869162471744831857869431, 10.15914639422395200692225500995, 10.37772506144100212875288646617, 10.99351839993430246414427951188
