| L(s) = 1 | + 2-s + 4-s + 8-s + 11-s + 2·13-s + 16-s + 6·17-s − 2·19-s + 22-s + 2·26-s + 6·29-s + 4·31-s + 32-s + 6·34-s − 2·37-s − 2·38-s + 6·41-s + 10·43-s + 44-s − 12·47-s + 2·52-s − 12·53-s + 6·58-s + 12·59-s + 10·61-s + 4·62-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 0.301·11-s + 0.554·13-s + 1/4·16-s + 1.45·17-s − 0.458·19-s + 0.213·22-s + 0.392·26-s + 1.11·29-s + 0.718·31-s + 0.176·32-s + 1.02·34-s − 0.328·37-s − 0.324·38-s + 0.937·41-s + 1.52·43-s + 0.150·44-s − 1.75·47-s + 0.277·52-s − 1.64·53-s + 0.787·58-s + 1.56·59-s + 1.28·61-s + 0.508·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.341159173\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.341159173\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−12.79185206082118, −12.47057874670569, −12.08635004213769, −11.46176877745313, −11.24303734987122, −10.55332949617661, −10.19425996563800, −9.724682270017510, −9.187665793196600, −8.574415367358250, −8.069510622357994, −7.771501983119005, −7.084267233466224, −6.585506271944061, −6.141756785325094, −5.739634875799317, −5.153251703770591, −4.580776140201027, −4.210466166596352, −3.393150175286877, −3.269400508803541, −2.481648407021661, −1.888855624734942, −1.126205479575044, −0.6784578020418052,
0.6784578020418052, 1.126205479575044, 1.888855624734942, 2.481648407021661, 3.269400508803541, 3.393150175286877, 4.210466166596352, 4.580776140201027, 5.153251703770591, 5.739634875799317, 6.141756785325094, 6.585506271944061, 7.084267233466224, 7.771501983119005, 8.069510622357994, 8.574415367358250, 9.187665793196600, 9.724682270017510, 10.19425996563800, 10.55332949617661, 11.24303734987122, 11.46176877745313, 12.08635004213769, 12.47057874670569, 12.79185206082118
