| L(s) = 1 | − 2-s − 2·3-s + 4-s + 5-s + 2·6-s − 7-s − 8-s + 9-s − 10-s + 2·11-s − 2·12-s + 14-s − 2·15-s + 16-s + 17-s − 18-s − 4·19-s + 20-s + 2·21-s − 2·22-s + 2·23-s + 2·24-s − 4·25-s + 4·27-s − 28-s − 5·29-s + 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s − 0.577·12-s + 0.267·14-s − 0.516·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.436·21-s − 0.426·22-s + 0.417·23-s + 0.408·24-s − 4/5·25-s + 0.769·27-s − 0.188·28-s − 0.928·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.646232968266110623892703978811, −7.88511284260259363518587850635, −6.77847554439076454621961190675, −6.36006863615857560170079985149, −5.69794086368469149240787920242, −4.79906348227654120324315885894, −3.67397356801609252210260633430, −2.43693973219870511221496064080, −1.25259749066607722047754053619, 0,
1.25259749066607722047754053619, 2.43693973219870511221496064080, 3.67397356801609252210260633430, 4.79906348227654120324315885894, 5.69794086368469149240787920242, 6.36006863615857560170079985149, 6.77847554439076454621961190675, 7.88511284260259363518587850635, 8.646232968266110623892703978811
