| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s − 36·6-s − 79·7-s + 64·8-s + 81·9-s + 150·11-s − 144·12-s + 137·13-s − 316·14-s + 256·16-s − 2.03e3·17-s + 324·18-s − 1.96e3·19-s + 711·21-s + 600·22-s − 1.35e3·23-s − 576·24-s + 548·26-s − 729·27-s − 1.26e3·28-s − 2.94e3·29-s + 713·31-s + 1.02e3·32-s − 1.35e3·33-s − 8.13e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.609·7-s + 0.353·8-s + 1/3·9-s + 0.373·11-s − 0.288·12-s + 0.224·13-s − 0.430·14-s + 1/4·16-s − 1.70·17-s + 0.235·18-s − 1.25·19-s + 0.351·21-s + 0.264·22-s − 0.532·23-s − 0.204·24-s + 0.158·26-s − 0.192·27-s − 0.304·28-s − 0.650·29-s + 0.133·31-s + 0.176·32-s − 0.215·33-s − 1.20·34-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 | \( 1 + 79 T + p^{5} T^{2} \) |
| 11 | \( 1 - 150 T + p^{5} T^{2} \) |
| 13 | \( 1 - 137 T + p^{5} T^{2} \) |
| 17 | \( 1 + 2034 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1969 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1350 T + p^{5} T^{2} \) |
| 29 | \( 1 + 2946 T + p^{5} T^{2} \) |
| 31 | \( 1 - 23 p T + p^{5} T^{2} \) |
| 37 | \( 1 + 3238 T + p^{5} T^{2} \) |
| 41 | \( 1 - 6564 T + p^{5} T^{2} \) |
| 43 | \( 1 + 19579 T + p^{5} T^{2} \) |
| 47 | \( 1 + 450 p T + p^{5} T^{2} \) |
| 53 | \( 1 + 25896 T + p^{5} T^{2} \) |
| 59 | \( 1 - 25350 T + p^{5} T^{2} \) |
| 61 | \( 1 - 50615 T + p^{5} T^{2} \) |
| 67 | \( 1 + 22519 T + p^{5} T^{2} \) |
| 71 | \( 1 - 33900 T + p^{5} T^{2} \) |
| 73 | \( 1 - 82442 T + p^{5} T^{2} \) |
| 79 | \( 1 + 81472 T + p^{5} T^{2} \) |
| 83 | \( 1 - 25782 T + p^{5} T^{2} \) |
| 89 | \( 1 - 103728 T + p^{5} T^{2} \) |
| 97 | \( 1 + 57343 T + p^{5} T^{2} \) |
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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
−11.60934655745817979986691117714, −10.88449317711071316145580181304, −9.725425848651348475424501639673, −8.418783582284956650134416219375, −6.78100891011517725087414603406, −6.24985290500022038634158597195, −4.82156841911186221131453214329, −3.72546720998941984813664065522, −2.02475002014909690000931529790, 0,
2.02475002014909690000931529790, 3.72546720998941984813664065522, 4.82156841911186221131453214329, 6.24985290500022038634158597195, 6.78100891011517725087414603406, 8.418783582284956650134416219375, 9.725425848651348475424501639673, 10.88449317711071316145580181304, 11.60934655745817979986691117714
