| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s + 12-s − 15-s + 16-s + 18-s + 2·19-s − 20-s − 12·23-s + 24-s − 2·25-s + 27-s − 9·29-s − 30-s + 32-s + 36-s + 2·38-s − 40-s − 14·43-s − 45-s − 12·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 0.258·15-s + 1/4·16-s + 0.235·18-s + 0.458·19-s − 0.223·20-s − 2.50·23-s + 0.204·24-s − 2/5·25-s + 0.192·27-s − 1.67·29-s − 0.182·30-s + 0.176·32-s + 1/6·36-s + 0.324·38-s − 0.158·40-s − 2.13·43-s − 0.149·45-s − 1.76·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 8 T + p T^{2} ) \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 140 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
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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
−8.098083757421628124373573879351, −7.913560268310654169476753169763, −7.46335730106859617827546934609, −6.89766586930783090179338216340, −6.44445446052109076941360171382, −5.93300642160155301599908297373, −5.47603056992212544648417436087, −4.92612974030375496961565795530, −4.30433285935080989623916938766, −3.88521559668010627685014100686, −3.43789265988245814699791501212, −2.93214368957060441967765204439, −1.94710384280292418352532319524, −1.71337707343189320894220678994, 0,
1.71337707343189320894220678994, 1.94710384280292418352532319524, 2.93214368957060441967765204439, 3.43789265988245814699791501212, 3.88521559668010627685014100686, 4.30433285935080989623916938766, 4.92612974030375496961565795530, 5.47603056992212544648417436087, 5.93300642160155301599908297373, 6.44445446052109076941360171382, 6.89766586930783090179338216340, 7.46335730106859617827546934609, 7.913560268310654169476753169763, 8.098083757421628124373573879351
