| L(s) = 1 | + 2.79·7-s − 3.10·11-s + 2.79·13-s − 4.83·17-s + 3.58·19-s − 4.83·23-s − 3.46·29-s + 6.58·31-s − 3.37·37-s − 6.56·41-s + 2.79·43-s + 8.66·47-s + 0.791·49-s + 4.83·53-s + 8.29·59-s + 10.3·61-s + 5·67-s + 7.93·71-s + 5·73-s − 8.66·77-s − 4·79-s − 1.00·83-s + 13.1·89-s + 7.79·91-s − 1.16·97-s + 15.2·101-s − 3.37·103-s + ⋯ |
| L(s) = 1 | + 1.05·7-s − 0.935·11-s + 0.774·13-s − 1.17·17-s + 0.821·19-s − 1.00·23-s − 0.643·29-s + 1.18·31-s − 0.554·37-s − 1.02·41-s + 0.425·43-s + 1.26·47-s + 0.113·49-s + 0.664·53-s + 1.08·59-s + 1.32·61-s + 0.610·67-s + 0.941·71-s + 0.585·73-s − 0.986·77-s − 0.450·79-s − 0.110·83-s + 1.39·89-s + 0.816·91-s − 0.118·97-s + 1.51·101-s − 0.332·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.131551423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.131551423\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2.79T + 7T^{2} \) |
| 11 | \( 1 + 3.10T + 11T^{2} \) |
| 13 | \( 1 - 2.79T + 13T^{2} \) |
| 17 | \( 1 + 4.83T + 17T^{2} \) |
| 19 | \( 1 - 3.58T + 19T^{2} \) |
| 23 | \( 1 + 4.83T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 - 6.58T + 31T^{2} \) |
| 37 | \( 1 + 3.37T + 37T^{2} \) |
| 41 | \( 1 + 6.56T + 41T^{2} \) |
| 43 | \( 1 - 2.79T + 43T^{2} \) |
| 47 | \( 1 - 8.66T + 47T^{2} \) |
| 53 | \( 1 - 4.83T + 53T^{2} \) |
| 59 | \( 1 - 8.29T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 5T + 67T^{2} \) |
| 71 | \( 1 - 7.93T + 71T^{2} \) |
| 73 | \( 1 - 5T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 1.00T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 + 1.16T + 97T^{2} \) |
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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
−7.976189393765721443599179556131, −7.18851175469456993129348189750, −6.48162319860747058169739431432, −5.58253710562728000675031460990, −5.11939822387148126084013133108, −4.29995306137915280329030198040, −3.60982681464726238118186452908, −2.49597360375491267286784153073, −1.86728320476343154805638531494, −0.72218947432561257461091312333,
0.72218947432561257461091312333, 1.86728320476343154805638531494, 2.49597360375491267286784153073, 3.60982681464726238118186452908, 4.29995306137915280329030198040, 5.11939822387148126084013133108, 5.58253710562728000675031460990, 6.48162319860747058169739431432, 7.18851175469456993129348189750, 7.976189393765721443599179556131
