| L(s) = 1 | − 2·3-s + 5-s + 4·7-s + 9-s + 11-s + 4·13-s − 2·15-s − 8·21-s + 6·23-s + 25-s + 4·27-s + 6·29-s + 8·31-s − 2·33-s + 4·35-s − 2·37-s − 8·39-s − 6·41-s − 8·43-s + 45-s − 6·47-s + 9·49-s + 6·53-s + 55-s − 12·59-s + 2·61-s + 4·63-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 0.516·15-s − 1.74·21-s + 1.25·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s + 1.43·31-s − 0.348·33-s + 0.676·35-s − 0.328·37-s − 1.28·39-s − 0.937·41-s − 1.21·43-s + 0.149·45-s − 0.875·47-s + 9/7·49-s + 0.824·53-s + 0.134·55-s − 1.56·59-s + 0.256·61-s + 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 317680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.80601828467949, −12.13755832103301, −11.85128502040321, −11.52213909506140, −11.10741133113431, −10.65592132118706, −10.31727984335906, −9.897767948191566, −8.997038809923081, −8.719711286537005, −8.333729890421925, −7.863143249731417, −7.127875708127540, −6.669828438668040, −6.284866992879903, −5.831327260672665, −5.219531274046290, −4.913639350321944, −4.536514240380294, −3.956804171869431, −3.008875280319147, −2.790493653542282, −1.629506664527557, −1.418366941518580, −0.9426221910729220, 0,
0.9426221910729220, 1.418366941518580, 1.629506664527557, 2.790493653542282, 3.008875280319147, 3.956804171869431, 4.536514240380294, 4.913639350321944, 5.219531274046290, 5.831327260672665, 6.284866992879903, 6.669828438668040, 7.127875708127540, 7.863143249731417, 8.333729890421925, 8.719711286537005, 8.997038809923081, 9.897767948191566, 10.31727984335906, 10.65592132118706, 11.10741133113431, 11.52213909506140, 11.85128502040321, 12.13755832103301, 12.80601828467949
