| L(s) = 1 | + 2·3-s + 62·7-s − 239·9-s − 144·11-s + 654·13-s + 1.19e3·17-s + 556·19-s + 124·21-s − 2.18e3·23-s − 964·27-s − 1.57e3·29-s + 9.66e3·31-s − 288·33-s + 3.53e3·37-s + 1.30e3·39-s + 7.46e3·41-s + 7.11e3·43-s + 2.82e4·47-s − 1.29e4·49-s + 2.38e3·51-s + 1.30e4·53-s + 1.11e3·57-s − 3.70e4·59-s + 3.95e4·61-s − 1.48e4·63-s + 5.67e4·67-s − 4.36e3·69-s + ⋯ |
| L(s) = 1 | + 0.128·3-s + 0.478·7-s − 0.983·9-s − 0.358·11-s + 1.07·13-s + 0.998·17-s + 0.353·19-s + 0.0613·21-s − 0.860·23-s − 0.254·27-s − 0.348·29-s + 1.80·31-s − 0.0460·33-s + 0.424·37-s + 0.137·39-s + 0.693·41-s + 0.586·43-s + 1.86·47-s − 0.771·49-s + 0.128·51-s + 0.637·53-s + 0.0453·57-s − 1.38·59-s + 1.36·61-s − 0.470·63-s + 1.54·67-s − 0.110·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.117312374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.117312374\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p^{5} T^{2} \) |
| 7 | \( 1 - 62 T + p^{5} T^{2} \) |
| 11 | \( 1 + 144 T + p^{5} T^{2} \) |
| 13 | \( 1 - 654 T + p^{5} T^{2} \) |
| 17 | \( 1 - 70 p T + p^{5} T^{2} \) |
| 19 | \( 1 - 556 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2182 T + p^{5} T^{2} \) |
| 29 | \( 1 + 1578 T + p^{5} T^{2} \) |
| 31 | \( 1 - 9660 T + p^{5} T^{2} \) |
| 37 | \( 1 - 3534 T + p^{5} T^{2} \) |
| 41 | \( 1 - 182 p T + p^{5} T^{2} \) |
| 43 | \( 1 - 7114 T + p^{5} T^{2} \) |
| 47 | \( 1 - 602 p T + p^{5} T^{2} \) |
| 53 | \( 1 - 13046 T + p^{5} T^{2} \) |
| 59 | \( 1 + 37092 T + p^{5} T^{2} \) |
| 61 | \( 1 - 39570 T + p^{5} T^{2} \) |
| 67 | \( 1 - 56734 T + p^{5} T^{2} \) |
| 71 | \( 1 - 45588 T + p^{5} T^{2} \) |
| 73 | \( 1 + 11842 T + p^{5} T^{2} \) |
| 79 | \( 1 - 94216 T + p^{5} T^{2} \) |
| 83 | \( 1 - 31482 T + p^{5} T^{2} \) |
| 89 | \( 1 + 94054 T + p^{5} T^{2} \) |
| 97 | \( 1 + 23714 T + p^{5} T^{2} \) |
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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
−11.53731584883971950463391096104, −10.71520137041825433442236739674, −9.577221298981009561459695844177, −8.406516697664370998512558876995, −7.79564809193443404650317311982, −6.22112944117083266453935973610, −5.34190101806751254587581256714, −3.85641369804727179356213633304, −2.56265465244481715687455353499, −0.919712787607494498226704685713,
0.919712787607494498226704685713, 2.56265465244481715687455353499, 3.85641369804727179356213633304, 5.34190101806751254587581256714, 6.22112944117083266453935973610, 7.79564809193443404650317311982, 8.406516697664370998512558876995, 9.577221298981009561459695844177, 10.71520137041825433442236739674, 11.53731584883971950463391096104
