| L(s) = 1 | − 6·3-s + 2·4-s + 62·5-s − 135·9-s + 22·11-s − 12·12-s − 372·15-s − 252·16-s + 124·20-s + 554·23-s + 1.63e3·25-s + 1.35e3·27-s − 2.72e3·31-s − 132·33-s − 270·36-s + 334·37-s + 44·44-s − 8.37e3·45-s + 3.40e3·47-s + 1.51e3·48-s + 1.80e3·49-s + 9.04e3·53-s + 1.36e3·55-s − 4.72e3·59-s − 744·60-s − 1.01e3·64-s − 5.60e3·67-s + ⋯ |
| L(s) = 1 | − 2/3·3-s + 1/8·4-s + 2.47·5-s − 5/3·9-s + 2/11·11-s − 0.0833·12-s − 1.65·15-s − 0.984·16-s + 0.309·20-s + 1.04·23-s + 2.61·25-s + 1.85·27-s − 2.83·31-s − 0.121·33-s − 0.208·36-s + 0.243·37-s + 1/44·44-s − 4.13·45-s + 1.54·47-s + 0.656·48-s + 0.750·49-s + 3.21·53-s + 0.450·55-s − 1.35·59-s − 0.206·60-s − 0.248·64-s − 1.24·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.157674621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.157674621\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_2$ | \( 1 - 2 p T + p^{4} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{8} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p T + p^{4} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 31 T + p^{4} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 1802 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22442 T^{2} + p^{8} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 114122 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 250922 T^{2} + p^{8} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 277 T + p^{4} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 38 p T + p^{4} T^{2} )( 1 + 38 p T + p^{4} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 1363 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 167 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 4522442 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 5385602 T^{2} + p^{8} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 1702 T + p^{4} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4522 T + p^{4} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 2363 T + p^{4} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 11966402 T^{2} + p^{8} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 2803 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3397 T + p^{4} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 45779402 T^{2} + p^{8} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 40803842 T^{2} + p^{8} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 94223522 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 4673 T + p^{4} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 4247 T + p^{4} T^{2} )^{2} \) |
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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
−20.26132514032608998620177529834, −19.76891243336154225506294972049, −18.23883335823624907732354596626, −18.18253336283652801360653841472, −17.34444638482209044977961468755, −16.87079131996468627874928504157, −16.56304172213857674847604833284, −15.07365855119871938737353357775, −14.28695245088176938995164735449, −13.74945259582314837992793239083, −13.16983060155429795253515143690, −12.05413188563647012617656866518, −11.05392227188633231096315632129, −10.52479981427678601601763710300, −9.204238464977936325510442516002, −8.991403689752017235867616356036, −6.90260280784055901568593389407, −5.73105080156715589507556740723, −5.53102188592883621196977418424, −2.36292077307959260973677058117,
2.36292077307959260973677058117, 5.53102188592883621196977418424, 5.73105080156715589507556740723, 6.90260280784055901568593389407, 8.991403689752017235867616356036, 9.204238464977936325510442516002, 10.52479981427678601601763710300, 11.05392227188633231096315632129, 12.05413188563647012617656866518, 13.16983060155429795253515143690, 13.74945259582314837992793239083, 14.28695245088176938995164735449, 15.07365855119871938737353357775, 16.56304172213857674847604833284, 16.87079131996468627874928504157, 17.34444638482209044977961468755, 18.18253336283652801360653841472, 18.23883335823624907732354596626, 19.76891243336154225506294972049, 20.26132514032608998620177529834
