| L(s) = 1 | + 2-s − 2·3-s − 2·4-s + 2·5-s − 2·6-s − 4·7-s − 3·8-s + 2·9-s + 2·10-s + 4·11-s + 4·12-s − 2·13-s − 4·14-s − 4·15-s + 16-s + 6·17-s + 2·18-s − 4·20-s + 8·21-s + 4·22-s − 2·23-s + 6·24-s − 7·25-s − 2·26-s − 6·27-s + 8·28-s + 10·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 4-s + 0.894·5-s − 0.816·6-s − 1.51·7-s − 1.06·8-s + 2/3·9-s + 0.632·10-s + 1.20·11-s + 1.15·12-s − 0.554·13-s − 1.06·14-s − 1.03·15-s + 1/4·16-s + 1.45·17-s + 0.471·18-s − 0.894·20-s + 1.74·21-s + 0.852·22-s − 0.417·23-s + 1.22·24-s − 7/5·25-s − 0.392·26-s − 1.15·27-s + 1.51·28-s + 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4492877238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4492877238\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 31 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 113 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 6 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 158 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 163 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−17.15432508999057702779671121261, −17.03606772532803878991628196034, −15.99097596480158213236404952174, −15.79402375241501110804903952077, −14.48246076460290596164391958315, −14.20506575973747679651369846082, −13.64320749624703022846928675733, −13.01060856769596994934602495263, −12.27803936712599513177270754436, −12.19411267640808601410552212031, −11.18435457248788706232861087885, −9.988884574157259088769712815632, −9.694681048392266970713798808449, −9.392764051520264102402757332269, −8.066873544053878311165268740079, −6.75174651250368878590349717473, −6.03278399652487520131783270143, −5.63181617197644712048767615573, −4.52821779104957415480228032986, −3.50551568171874308730107016287,
3.50551568171874308730107016287, 4.52821779104957415480228032986, 5.63181617197644712048767615573, 6.03278399652487520131783270143, 6.75174651250368878590349717473, 8.066873544053878311165268740079, 9.392764051520264102402757332269, 9.694681048392266970713798808449, 9.988884574157259088769712815632, 11.18435457248788706232861087885, 12.19411267640808601410552212031, 12.27803936712599513177270754436, 13.01060856769596994934602495263, 13.64320749624703022846928675733, 14.20506575973747679651369846082, 14.48246076460290596164391958315, 15.79402375241501110804903952077, 15.99097596480158213236404952174, 17.03606772532803878991628196034, 17.15432508999057702779671121261
