| L(s) = 1 | − 2·4-s − 6·7-s − 2·13-s + 4·16-s − 6·19-s − 10·25-s + 12·28-s + 10·37-s + 17·49-s + 4·52-s − 14·61-s − 8·64-s − 6·67-s + 10·73-s + 12·76-s + 30·79-s + 12·91-s − 14·97-s + 20·100-s + 101-s + 103-s + 107-s + 109-s − 24·112-s + 113-s − 22·121-s + 127-s + ⋯ |
| L(s) = 1 | − 4-s − 2.26·7-s − 0.554·13-s + 16-s − 1.37·19-s − 2·25-s + 2.26·28-s + 1.64·37-s + 17/7·49-s + 0.554·52-s − 1.79·61-s − 64-s − 0.733·67-s + 1.17·73-s + 1.37·76-s + 3.37·79-s + 1.25·91-s − 1.42·97-s + 2·100-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s − 2.26·112-s + 0.0940·113-s − 2·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−16.6317848406, −16.3000171353, −15.6530617256, −15.0864369573, −14.8921107674, −13.9619381519, −13.5821136983, −13.2322338022, −12.7156394907, −12.3637426331, −11.8905824235, −10.9087282930, −10.4015015465, −9.77257431826, −9.42919920821, −9.22003132460, −8.21765036746, −7.78745335695, −6.91823062220, −6.08584189960, −6.04893540001, −4.87832652993, −4.04304401380, −3.51757494847, −2.49929826070, 0,
2.49929826070, 3.51757494847, 4.04304401380, 4.87832652993, 6.04893540001, 6.08584189960, 6.91823062220, 7.78745335695, 8.21765036746, 9.22003132460, 9.42919920821, 9.77257431826, 10.4015015465, 10.9087282930, 11.8905824235, 12.3637426331, 12.7156394907, 13.2322338022, 13.5821136983, 13.9619381519, 14.8921107674, 15.0864369573, 15.6530617256, 16.3000171353, 16.6317848406
