L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s − 36·6-s − 7-s − 64·8-s + 81·9-s − 210·11-s + 144·12-s − 667·13-s + 4·14-s + 256·16-s + 114·17-s − 324·18-s + 581·19-s − 9·21-s + 840·22-s − 4.35e3·23-s − 576·24-s + 2.66e3·26-s + 729·27-s − 16·28-s − 126·29-s + 7.58e3·31-s − 1.02e3·32-s − 1.89e3·33-s − 456·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.00771·7-s − 0.353·8-s + 1/3·9-s − 0.523·11-s + 0.288·12-s − 1.09·13-s + 0.00545·14-s + 1/4·16-s + 0.0956·17-s − 0.235·18-s + 0.369·19-s − 0.00445·21-s + 0.370·22-s − 1.71·23-s − 0.204·24-s + 0.774·26-s + 0.192·27-s − 0.00385·28-s − 0.0278·29-s + 1.41·31-s − 0.176·32-s − 0.302·33-s − 0.0676·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 - p^{2} T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + T + p^{5} T^{2} \) |
| 11 | \( 1 + 210 T + p^{5} T^{2} \) |
| 13 | \( 1 + 667 T + p^{5} T^{2} \) |
| 17 | \( 1 - 114 T + p^{5} T^{2} \) |
| 19 | \( 1 - 581 T + p^{5} T^{2} \) |
| 23 | \( 1 + 4350 T + p^{5} T^{2} \) |
| 29 | \( 1 + 126 T + p^{5} T^{2} \) |
| 31 | \( 1 - 7583 T + p^{5} T^{2} \) |
| 37 | \( 1 + 3742 T + p^{5} T^{2} \) |
| 41 | \( 1 + 2856 T + p^{5} T^{2} \) |
| 43 | \( 1 + 18241 T + p^{5} T^{2} \) |
| 47 | \( 1 + 23370 T + p^{5} T^{2} \) |
| 53 | \( 1 + 21684 T + p^{5} T^{2} \) |
| 59 | \( 1 + 32310 T + p^{5} T^{2} \) |
| 61 | \( 1 + 7165 T + p^{5} T^{2} \) |
| 67 | \( 1 - 59579 T + p^{5} T^{2} \) |
| 71 | \( 1 + 43080 T + p^{5} T^{2} \) |
| 73 | \( 1 + 28942 T + p^{5} T^{2} \) |
| 79 | \( 1 - 27608 T + p^{5} T^{2} \) |
| 83 | \( 1 + 1782 T + p^{5} T^{2} \) |
| 89 | \( 1 - 50208 T + p^{5} T^{2} \) |
| 97 | \( 1 - 142793 T + p^{5} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.58367188932704811794021752768, −10.16223086079876499335452588070, −9.700501931417987180744364090900, −8.336720911792825821994889627327, −7.67445350059577524699688081466, −6.42637810910469364890500434719, −4.84634474047403086297118170014, −3.13665459306655040049505795703, −1.86229455487229835818126326278, 0,
1.86229455487229835818126326278, 3.13665459306655040049505795703, 4.84634474047403086297118170014, 6.42637810910469364890500434719, 7.67445350059577524699688081466, 8.336720911792825821994889627327, 9.700501931417987180744364090900, 10.16223086079876499335452588070, 11.58367188932704811794021752768