| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 6·6-s + 4·7-s − 8·8-s + 9·9-s − 48·11-s − 12·12-s − 2·13-s − 8·14-s + 16·16-s + 114·17-s − 18·18-s + 140·19-s − 12·21-s + 96·22-s − 72·23-s + 24·24-s + 4·26-s − 27·27-s + 16·28-s + 210·29-s + 272·31-s − 32·32-s + 144·33-s − 228·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.215·7-s − 0.353·8-s + 1/3·9-s − 1.31·11-s − 0.288·12-s − 0.0426·13-s − 0.152·14-s + 1/4·16-s + 1.62·17-s − 0.235·18-s + 1.69·19-s − 0.124·21-s + 0.930·22-s − 0.652·23-s + 0.204·24-s + 0.0301·26-s − 0.192·27-s + 0.107·28-s + 1.34·29-s + 1.57·31-s − 0.176·32-s + 0.759·33-s − 1.15·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9728532543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9728532543\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 48 T + p^{3} T^{2} \) |
| 13 | \( 1 + 2 T + p^{3} T^{2} \) |
| 17 | \( 1 - 114 T + p^{3} T^{2} \) |
| 19 | \( 1 - 140 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 - 210 T + p^{3} T^{2} \) |
| 31 | \( 1 - 272 T + p^{3} T^{2} \) |
| 37 | \( 1 - 334 T + p^{3} T^{2} \) |
| 41 | \( 1 + 198 T + p^{3} T^{2} \) |
| 43 | \( 1 - 268 T + p^{3} T^{2} \) |
| 47 | \( 1 + 216 T + p^{3} T^{2} \) |
| 53 | \( 1 - 78 T + p^{3} T^{2} \) |
| 59 | \( 1 - 240 T + p^{3} T^{2} \) |
| 61 | \( 1 - 302 T + p^{3} T^{2} \) |
| 67 | \( 1 + 596 T + p^{3} T^{2} \) |
| 71 | \( 1 + 768 T + p^{3} T^{2} \) |
| 73 | \( 1 - 478 T + p^{3} T^{2} \) |
| 79 | \( 1 + 640 T + p^{3} T^{2} \) |
| 83 | \( 1 - 348 T + p^{3} T^{2} \) |
| 89 | \( 1 - 210 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1534 T + p^{3} T^{2} \) |
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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
−12.20758137677265803104307715077, −11.55095394237526904523979876871, −10.26898663844494974240945581273, −9.824067993069609445006730935917, −8.156247520841036838033404597455, −7.51284248276558635512285968019, −6.02899418834989501059491386147, −4.97764266964171710255422014039, −2.93604958615833487260583856938, −0.949258137058925379836801763546,
0.949258137058925379836801763546, 2.93604958615833487260583856938, 4.97764266964171710255422014039, 6.02899418834989501059491386147, 7.51284248276558635512285968019, 8.156247520841036838033404597455, 9.824067993069609445006730935917, 10.26898663844494974240945581273, 11.55095394237526904523979876871, 12.20758137677265803104307715077
