| 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
| \(L(3)\) |
\(\approx\) |
\(2.631954432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.631954432\) |
| \(L(\frac{7}{2})\) |
|
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
−12.15553845231285307541850331027, −11.16047832932594305991834521737, −10.47178045872736596719365799553, −9.020809415820747954732763893721, −7.64414786838266969356564741357, −6.47932332874867544507572303291, −5.48776814719245394986454170828, −4.35713328202806311832611237939, −2.90345403548647084071504445069, −1.05591794185310026990596328749,
1.05591794185310026990596328749, 2.90345403548647084071504445069, 4.35713328202806311832611237939, 5.48776814719245394986454170828, 6.47932332874867544507572303291, 7.64414786838266969356564741357, 9.020809415820747954732763893721, 10.47178045872736596719365799553, 11.16047832932594305991834521737, 12.15553845231285307541850331027
