| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 12-s + 16-s − 18-s − 2·19-s + 23-s − 24-s − 4·25-s + 27-s − 10·29-s − 32-s + 36-s + 2·38-s − 3·43-s − 46-s − 12·47-s + 48-s + 49-s + 4·50-s + 4·53-s − 54-s − 2·57-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 1/4·16-s − 0.235·18-s − 0.458·19-s + 0.208·23-s − 0.204·24-s − 4/5·25-s + 0.192·27-s − 1.85·29-s − 0.176·32-s + 1/6·36-s + 0.324·38-s − 0.457·43-s − 0.147·46-s − 1.75·47-s + 0.144·48-s + 1/7·49-s + 0.565·50-s + 0.549·53-s − 0.136·54-s − 0.264·57-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$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 9 T + p T^{2} ) \) |
| good | 7 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.352787152315599255463708448843, −7.919844981851994033455286235572, −7.48905722650298899334882411140, −7.05344638473707413635400485777, −6.55758958522201984737707805424, −6.14818226522775773129178267125, −5.44267556463449754798083530785, −5.12601240672620755785374191286, −4.31755794231981864165348702655, −3.73639241170596922999195000828, −3.38994154586335111339876996087, −2.51357983800067797382050303273, −2.04165340343225427517473555984, −1.32628334621379393024094532167, 0,
1.32628334621379393024094532167, 2.04165340343225427517473555984, 2.51357983800067797382050303273, 3.38994154586335111339876996087, 3.73639241170596922999195000828, 4.31755794231981864165348702655, 5.12601240672620755785374191286, 5.44267556463449754798083530785, 6.14818226522775773129178267125, 6.55758958522201984737707805424, 7.05344638473707413635400485777, 7.48905722650298899334882411140, 7.919844981851994033455286235572, 8.352787152315599255463708448843
