| L(s) = 1 | + 3-s − 2·4-s − 5-s + 9-s − 2·12-s − 2·13-s − 15-s + 4·16-s + 2·20-s + 25-s + 27-s − 8·31-s − 2·36-s + 10·37-s − 2·39-s − 18·41-s + 4·43-s − 45-s + 4·48-s + 2·49-s + 4·52-s − 12·53-s + 2·60-s − 8·64-s + 2·65-s − 8·67-s + 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s + 1/3·9-s − 0.577·12-s − 0.554·13-s − 0.258·15-s + 16-s + 0.447·20-s + 1/5·25-s + 0.192·27-s − 1.43·31-s − 1/3·36-s + 1.64·37-s − 0.320·39-s − 2.81·41-s + 0.609·43-s − 0.149·45-s + 0.577·48-s + 2/7·49-s + 0.554·52-s − 1.64·53-s + 0.258·60-s − 64-s + 0.248·65-s − 0.977·67-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 170 T^{2} + p^{2} T^{4} \) |
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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
−8.639234930650642765620751129680, −8.523775257728889674307296197910, −7.895646387548021642457581481954, −7.43218262021209478220184602720, −7.17125924843781199120084644241, −6.32168167241113410404567831150, −5.88169747075832064862554687099, −5.10907947613849939441179482998, −4.77290465581652273268316475331, −4.24331545329566705598520649279, −3.54614090615107725449531004688, −3.21991610284210338990303496611, −2.30332128407427443161963864401, −1.36328580534491248673091037460, 0,
1.36328580534491248673091037460, 2.30332128407427443161963864401, 3.21991610284210338990303496611, 3.54614090615107725449531004688, 4.24331545329566705598520649279, 4.77290465581652273268316475331, 5.10907947613849939441179482998, 5.88169747075832064862554687099, 6.32168167241113410404567831150, 7.17125924843781199120084644241, 7.43218262021209478220184602720, 7.895646387548021642457581481954, 8.523775257728889674307296197910, 8.639234930650642765620751129680
