| L(s) = 1 | + 3·2-s + 3·3-s + 4-s + 9·6-s − 7·7-s − 21·8-s + 9·9-s − 6·11-s + 3·12-s − 41·13-s − 21·14-s − 71·16-s − 27·17-s + 27·18-s − 4·19-s − 21·21-s − 18·22-s − 75·23-s − 63·24-s − 123·26-s + 27·27-s − 7·28-s − 123·29-s − 205·31-s − 45·32-s − 18·33-s − 81·34-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 0.577·3-s + 1/8·4-s + 0.612·6-s − 0.377·7-s − 0.928·8-s + 1/3·9-s − 0.164·11-s + 0.0721·12-s − 0.874·13-s − 0.400·14-s − 1.10·16-s − 0.385·17-s + 0.353·18-s − 0.0482·19-s − 0.218·21-s − 0.174·22-s − 0.679·23-s − 0.535·24-s − 0.927·26-s + 0.192·27-s − 0.0472·28-s − 0.787·29-s − 1.18·31-s − 0.248·32-s − 0.0949·33-s − 0.408·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 11 | \( 1 + 6 T + p^{3} T^{2} \) |
| 13 | \( 1 + 41 T + p^{3} T^{2} \) |
| 17 | \( 1 + 27 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 T + p^{3} T^{2} \) |
| 23 | \( 1 + 75 T + p^{3} T^{2} \) |
| 29 | \( 1 + 123 T + p^{3} T^{2} \) |
| 31 | \( 1 + 205 T + p^{3} T^{2} \) |
| 37 | \( 1 - 262 T + p^{3} T^{2} \) |
| 41 | \( 1 - 57 T + p^{3} T^{2} \) |
| 43 | \( 1 + 407 T + p^{3} T^{2} \) |
| 47 | \( 1 - 60 T + p^{3} T^{2} \) |
| 53 | \( 1 + 327 T + p^{3} T^{2} \) |
| 59 | \( 1 - 33 T + p^{3} T^{2} \) |
| 61 | \( 1 + 7 p T + p^{3} T^{2} \) |
| 67 | \( 1 - 628 T + p^{3} T^{2} \) |
| 71 | \( 1 - 300 T + p^{3} T^{2} \) |
| 73 | \( 1 + 98 T + p^{3} T^{2} \) |
| 79 | \( 1 - 686 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1401 T + p^{3} T^{2} \) |
| 89 | \( 1 - 714 T + p^{3} T^{2} \) |
| 97 | \( 1 + 494 T + p^{3} T^{2} \) |
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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
−9.807302045796272295674898808054, −9.257496859134056000011165989452, −8.188394656051961823207356061331, −7.15749955504401947900929131498, −6.12558197363421899011780380839, −5.12353152151255960979168735074, −4.17161008577882404777006961996, −3.24748370261333307457715623213, −2.17896913760619259700923150123, 0,
2.17896913760619259700923150123, 3.24748370261333307457715623213, 4.17161008577882404777006961996, 5.12353152151255960979168735074, 6.12558197363421899011780380839, 7.15749955504401947900929131498, 8.188394656051961823207356061331, 9.257496859134056000011165989452, 9.807302045796272295674898808054
