| L(s) = 1 | + 3-s − 2.37·5-s − 7-s + 9-s − 3.37·11-s − 0.372·13-s − 2.37·15-s + 4·17-s − 7.37·19-s − 21-s − 23-s + 0.627·25-s + 27-s − 0.372·29-s + 6·31-s − 3.37·33-s + 2.37·35-s − 8.37·37-s − 0.372·39-s + 5.74·41-s + 2.37·43-s − 2.37·45-s + 7.74·47-s + 49-s + 4·51-s − 10.1·53-s + 8·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.06·5-s − 0.377·7-s + 0.333·9-s − 1.01·11-s − 0.103·13-s − 0.612·15-s + 0.970·17-s − 1.69·19-s − 0.218·21-s − 0.208·23-s + 0.125·25-s + 0.192·27-s − 0.0691·29-s + 1.07·31-s − 0.587·33-s + 0.400·35-s − 1.37·37-s − 0.0596·39-s + 0.897·41-s + 0.361·43-s − 0.353·45-s + 1.12·47-s + 0.142·49-s + 0.560·51-s − 1.38·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.209473298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.209473298\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 2.37T + 5T^{2} \) |
| 11 | \( 1 + 3.37T + 11T^{2} \) |
| 13 | \( 1 + 0.372T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 7.37T + 19T^{2} \) |
| 29 | \( 1 + 0.372T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 8.37T + 37T^{2} \) |
| 41 | \( 1 - 5.74T + 41T^{2} \) |
| 43 | \( 1 - 2.37T + 43T^{2} \) |
| 47 | \( 1 - 7.74T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 9.37T + 59T^{2} \) |
| 61 | \( 1 + 2.62T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 12T + 73T^{2} \) |
| 79 | \( 1 - 2.74T + 79T^{2} \) |
| 83 | \( 1 + 2.74T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 5.11T + 97T^{2} \) |
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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
−7.80406127815212048084138515442, −7.48600622705977314119406057216, −6.54007521929491323703797781594, −5.85344377678486701385229915042, −4.85399504610240050414086337419, −4.23007640608338524492251484060, −3.48541038078183103078959796312, −2.81754412317450386034233329442, −1.93928301331908762596962685366, −0.50953969450420984073089415385,
0.50953969450420984073089415385, 1.93928301331908762596962685366, 2.81754412317450386034233329442, 3.48541038078183103078959796312, 4.23007640608338524492251484060, 4.85399504610240050414086337419, 5.85344377678486701385229915042, 6.54007521929491323703797781594, 7.48600622705977314119406057216, 7.80406127815212048084138515442
