| L(s) = 1 | + 2-s − 3·3-s − 7·4-s − 10·5-s − 3·6-s − 8·7-s − 15·8-s + 9·9-s − 10·10-s + 12·11-s + 21·12-s − 26·13-s − 8·14-s + 30·15-s + 41·16-s + 17·17-s + 9·18-s − 148·19-s + 70·20-s + 24·21-s + 12·22-s + 152·23-s + 45·24-s − 25·25-s − 26·26-s − 27·27-s + 56·28-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 0.577·3-s − 7/8·4-s − 0.894·5-s − 0.204·6-s − 0.431·7-s − 0.662·8-s + 1/3·9-s − 0.316·10-s + 0.328·11-s + 0.505·12-s − 0.554·13-s − 0.152·14-s + 0.516·15-s + 0.640·16-s + 0.242·17-s + 0.117·18-s − 1.78·19-s + 0.782·20-s + 0.249·21-s + 0.116·22-s + 1.37·23-s + 0.382·24-s − 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 17 | \( 1 - p T \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 5 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 7 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 19 | \( 1 + 148 T + p^{3} T^{2} \) |
| 23 | \( 1 - 152 T + p^{3} T^{2} \) |
| 29 | \( 1 + 66 T + p^{3} T^{2} \) |
| 31 | \( 1 + 32 T + p^{3} T^{2} \) |
| 37 | \( 1 + 266 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 T + p^{3} T^{2} \) |
| 43 | \( 1 + 92 T + p^{3} T^{2} \) |
| 47 | \( 1 + 288 T + p^{3} T^{2} \) |
| 53 | \( 1 + 546 T + p^{3} T^{2} \) |
| 59 | \( 1 - 420 T + p^{3} T^{2} \) |
| 61 | \( 1 - 350 T + p^{3} T^{2} \) |
| 67 | \( 1 - 940 T + p^{3} T^{2} \) |
| 71 | \( 1 - 424 T + p^{3} T^{2} \) |
| 73 | \( 1 - 378 T + p^{3} T^{2} \) |
| 79 | \( 1 - 288 T + p^{3} T^{2} \) |
| 83 | \( 1 - 748 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1558 T + p^{3} T^{2} \) |
| 97 | \( 1 - 530 T + p^{3} T^{2} \) |
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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
−14.53904281689073804001849152507, −13.03347838368574969541591465291, −12.34642122972081777803581255646, −11.08693016308593066975550004235, −9.646955761799477396510989148532, −8.317446370554701286463420870181, −6.68359733601792221802316329844, −5.02818678526402075597091196574, −3.74899332871973191320188228928, 0,
3.74899332871973191320188228928, 5.02818678526402075597091196574, 6.68359733601792221802316329844, 8.317446370554701286463420870181, 9.646955761799477396510989148532, 11.08693016308593066975550004235, 12.34642122972081777803581255646, 13.03347838368574969541591465291, 14.53904281689073804001849152507
