| L(s) = 1 | + 2·3-s + 4·5-s + 3·9-s + 4·11-s + 8·15-s − 4·17-s + 8·19-s + 4·23-s + 4·25-s + 4·27-s + 16·31-s + 8·33-s − 8·37-s − 4·41-s − 16·43-s + 12·45-s + 8·47-s − 8·51-s − 4·53-s + 16·55-s + 16·57-s + 16·59-s + 8·61-s − 16·67-s + 8·69-s + 4·71-s − 8·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s + 9-s + 1.20·11-s + 2.06·15-s − 0.970·17-s + 1.83·19-s + 0.834·23-s + 4/5·25-s + 0.769·27-s + 2.87·31-s + 1.39·33-s − 1.31·37-s − 0.624·41-s − 2.43·43-s + 1.78·45-s + 1.16·47-s − 1.12·51-s − 0.549·53-s + 2.15·55-s + 2.11·57-s + 2.08·59-s + 1.02·61-s − 1.95·67-s + 0.963·69-s + 0.474·71-s − 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.095176314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.095176314\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 16 T + 118 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 84 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 112 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 24 T + 320 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
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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
−9.861077245890059112473975054621, −9.708626600415256576266509974761, −8.988043505261851823510339518785, −8.923978473591407282610553579757, −8.544660440433762910139807674980, −8.101603910814072625241201013756, −7.43237885148402138107504187746, −7.03168012505967240256424756390, −6.53073269406908781822797002563, −6.51248393570861666948610462271, −5.67220738639257175594890516254, −5.43706643098702948040279093043, −4.65139888895648503476376301733, −4.54842552465306401078625023296, −3.52549578331878025912284453761, −3.37231425399933199119704063175, −2.60595467631557241019258042097, −2.25556487916664800711073439391, −1.47687880858686783741850245008, −1.18230171322547453703821044986,
1.18230171322547453703821044986, 1.47687880858686783741850245008, 2.25556487916664800711073439391, 2.60595467631557241019258042097, 3.37231425399933199119704063175, 3.52549578331878025912284453761, 4.54842552465306401078625023296, 4.65139888895648503476376301733, 5.43706643098702948040279093043, 5.67220738639257175594890516254, 6.51248393570861666948610462271, 6.53073269406908781822797002563, 7.03168012505967240256424756390, 7.43237885148402138107504187746, 8.101603910814072625241201013756, 8.544660440433762910139807674980, 8.923978473591407282610553579757, 8.988043505261851823510339518785, 9.708626600415256576266509974761, 9.861077245890059112473975054621
