| L(s) = 1 | + 7.99·3-s + 5·5-s − 9.93·7-s + 36.9·9-s − 60.0·11-s + 32.3·13-s + 39.9·15-s + 12.5·17-s − 116.·19-s − 79.4·21-s − 204.·23-s + 25·25-s + 79.7·27-s − 31.8·29-s − 81.0·31-s − 480.·33-s − 49.6·35-s + 270.·37-s + 258.·39-s − 330.·41-s − 138.·43-s + 184.·45-s + 211.·47-s − 244.·49-s + 100.·51-s + 437.·53-s − 300.·55-s + ⋯ |
| L(s) = 1 | + 1.53·3-s + 0.447·5-s − 0.536·7-s + 1.36·9-s − 1.64·11-s + 0.689·13-s + 0.688·15-s + 0.178·17-s − 1.41·19-s − 0.825·21-s − 1.85·23-s + 0.200·25-s + 0.568·27-s − 0.203·29-s − 0.469·31-s − 2.53·33-s − 0.239·35-s + 1.20·37-s + 1.06·39-s − 1.25·41-s − 0.490·43-s + 0.612·45-s + 0.656·47-s − 0.712·49-s + 0.275·51-s + 1.13·53-s − 0.735·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 3 | \( 1 - 7.99T + 27T^{2} \) |
| 7 | \( 1 + 9.93T + 343T^{2} \) |
| 11 | \( 1 + 60.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 32.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 12.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 204.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 31.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 81.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 270.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 330.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 138.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 211.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 437.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 181.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 472.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 71.8T + 3.00e5T^{2} \) |
| 71 | \( 1 - 767.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 261.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 600.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 612.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 503.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 500.T + 9.12e5T^{2} \) |
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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
−8.730595572778350580632557505498, −8.200986196244737107243908700558, −7.57316100764805992063501550368, −6.43026726681854179409219500914, −5.60970228448851984659187746142, −4.32459584596057889307935277503, −3.43611723294409045982209927708, −2.53555095235012548279964992415, −1.86601856819716245500367849479, 0,
1.86601856819716245500367849479, 2.53555095235012548279964992415, 3.43611723294409045982209927708, 4.32459584596057889307935277503, 5.60970228448851984659187746142, 6.43026726681854179409219500914, 7.57316100764805992063501550368, 8.200986196244737107243908700558, 8.730595572778350580632557505498
