| L(s) = 1 | + 2-s + 4-s + 8-s + 6·11-s − 13-s + 16-s + 6·19-s + 6·22-s + 6·23-s − 26-s − 2·29-s + 4·31-s + 32-s − 10·37-s + 6·38-s + 6·41-s − 8·43-s + 6·44-s + 6·46-s + 8·47-s − 7·49-s − 52-s − 6·53-s − 2·58-s − 10·59-s − 6·61-s + 4·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.80·11-s − 0.277·13-s + 1/4·16-s + 1.37·19-s + 1.27·22-s + 1.25·23-s − 0.196·26-s − 0.371·29-s + 0.718·31-s + 0.176·32-s − 1.64·37-s + 0.973·38-s + 0.937·41-s − 1.21·43-s + 0.904·44-s + 0.884·46-s + 1.16·47-s − 49-s − 0.138·52-s − 0.824·53-s − 0.262·58-s − 1.30·59-s − 0.768·61-s + 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.902962065\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.902962065\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−7.921525232481233455836187644508, −7.20295440422604640679362774429, −6.64137188592535970839958666193, −5.99777636214939258464949374309, −5.08311424327071635909807948641, −4.55162559664148375312828966593, −3.53977849542417401891160440531, −3.13755730419709917810629840049, −1.84571826141549283370076327881, −1.01518726291618464357056706942,
1.01518726291618464357056706942, 1.84571826141549283370076327881, 3.13755730419709917810629840049, 3.53977849542417401891160440531, 4.55162559664148375312828966593, 5.08311424327071635909807948641, 5.99777636214939258464949374309, 6.64137188592535970839958666193, 7.20295440422604640679362774429, 7.921525232481233455836187644508
