| L(s) = 1 | + 1.36·3-s − 2.47·7-s − 1.12·9-s − 0.151·11-s − 6.69·13-s + 5.77·17-s − 1.64·19-s − 3.39·21-s + 1.12·23-s − 5.64·27-s − 0.332·29-s + 2.82·31-s − 0.206·33-s + 2.96·37-s − 9.16·39-s − 7.82·41-s − 6.46·43-s + 4.10·47-s − 0.866·49-s + 7.90·51-s + 5.62·53-s − 2.25·57-s + 7.98·59-s + 1.49·61-s + 2.78·63-s + 7.44·67-s + 1.54·69-s + ⋯ |
| L(s) = 1 | + 0.790·3-s − 0.936·7-s − 0.374·9-s − 0.0455·11-s − 1.85·13-s + 1.40·17-s − 0.377·19-s − 0.740·21-s + 0.235·23-s − 1.08·27-s − 0.0617·29-s + 0.506·31-s − 0.0360·33-s + 0.488·37-s − 1.46·39-s − 1.22·41-s − 0.985·43-s + 0.598·47-s − 0.123·49-s + 1.10·51-s + 0.772·53-s − 0.298·57-s + 1.03·59-s + 0.191·61-s + 0.351·63-s + 0.909·67-s + 0.185·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.596424749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.596424749\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 1.36T + 3T^{2} \) |
| 7 | \( 1 + 2.47T + 7T^{2} \) |
| 11 | \( 1 + 0.151T + 11T^{2} \) |
| 13 | \( 1 + 6.69T + 13T^{2} \) |
| 17 | \( 1 - 5.77T + 17T^{2} \) |
| 19 | \( 1 + 1.64T + 19T^{2} \) |
| 23 | \( 1 - 1.12T + 23T^{2} \) |
| 29 | \( 1 + 0.332T + 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 - 2.96T + 37T^{2} \) |
| 41 | \( 1 + 7.82T + 41T^{2} \) |
| 43 | \( 1 + 6.46T + 43T^{2} \) |
| 47 | \( 1 - 4.10T + 47T^{2} \) |
| 53 | \( 1 - 5.62T + 53T^{2} \) |
| 59 | \( 1 - 7.98T + 59T^{2} \) |
| 61 | \( 1 - 1.49T + 61T^{2} \) |
| 67 | \( 1 - 7.44T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 - 3.44T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 - 17.5T + 83T^{2} \) |
| 89 | \( 1 - 9.35T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−7.79952586557543839619490881336, −7.01054635422406144301432580196, −6.45144289018896079655469967325, −5.48390503334945631796378922028, −5.03517849900308923746498371459, −3.96958418562215522329438431712, −3.23904304973891834071753224452, −2.72713025372001427895073066921, −1.98623139934804872476035743442, −0.54049260235784219811418418690,
0.54049260235784219811418418690, 1.98623139934804872476035743442, 2.72713025372001427895073066921, 3.23904304973891834071753224452, 3.96958418562215522329438431712, 5.03517849900308923746498371459, 5.48390503334945631796378922028, 6.45144289018896079655469967325, 7.01054635422406144301432580196, 7.79952586557543839619490881336
