| L(s) = 1 | + 1.79·3-s + 2.90·5-s + 4.51·7-s + 0.229·9-s − 3.50·11-s + 3.43·13-s + 5.22·15-s + 5.02·17-s + 0.0967·19-s + 8.11·21-s + 2.63·23-s + 3.46·25-s − 4.97·27-s − 1.78·29-s + 0.0155·31-s − 6.30·33-s + 13.1·35-s − 1.64·37-s + 6.16·39-s + 2.90·41-s − 5.59·43-s + 0.667·45-s − 12.4·47-s + 13.3·49-s + 9.03·51-s − 3.37·53-s − 10.2·55-s + ⋯ |
| L(s) = 1 | + 1.03·3-s + 1.30·5-s + 1.70·7-s + 0.0764·9-s − 1.05·11-s + 0.951·13-s + 1.34·15-s + 1.21·17-s + 0.0221·19-s + 1.76·21-s + 0.549·23-s + 0.692·25-s − 0.958·27-s − 0.331·29-s + 0.00278·31-s − 1.09·33-s + 2.21·35-s − 0.270·37-s + 0.987·39-s + 0.454·41-s − 0.852·43-s + 0.0995·45-s − 1.81·47-s + 1.90·49-s + 1.26·51-s − 0.464·53-s − 1.37·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.400618097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.400618097\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 1.79T + 3T^{2} \) |
| 5 | \( 1 - 2.90T + 5T^{2} \) |
| 7 | \( 1 - 4.51T + 7T^{2} \) |
| 11 | \( 1 + 3.50T + 11T^{2} \) |
| 13 | \( 1 - 3.43T + 13T^{2} \) |
| 17 | \( 1 - 5.02T + 17T^{2} \) |
| 19 | \( 1 - 0.0967T + 19T^{2} \) |
| 23 | \( 1 - 2.63T + 23T^{2} \) |
| 29 | \( 1 + 1.78T + 29T^{2} \) |
| 31 | \( 1 - 0.0155T + 31T^{2} \) |
| 37 | \( 1 + 1.64T + 37T^{2} \) |
| 41 | \( 1 - 2.90T + 41T^{2} \) |
| 43 | \( 1 + 5.59T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 3.37T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 0.777T + 61T^{2} \) |
| 67 | \( 1 - 6.11T + 67T^{2} \) |
| 71 | \( 1 - 3.72T + 71T^{2} \) |
| 73 | \( 1 + 7.62T + 73T^{2} \) |
| 79 | \( 1 + 14.3T + 79T^{2} \) |
| 83 | \( 1 + 0.694T + 83T^{2} \) |
| 89 | \( 1 - 0.821T + 89T^{2} \) |
| 97 | \( 1 + 0.292T + 97T^{2} \) |
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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
−8.336087291148535323846459624034, −8.011556403336425792757951435732, −7.22346333189074729301208498018, −6.02967190296508779603759044043, −5.40350773785860839905955514676, −4.88804286735496800274981615411, −3.65383644129654370605005942479, −2.79433866004236352417938509804, −1.96549687439444697345264909501, −1.33027667330455384010270679409,
1.33027667330455384010270679409, 1.96549687439444697345264909501, 2.79433866004236352417938509804, 3.65383644129654370605005942479, 4.88804286735496800274981615411, 5.40350773785860839905955514676, 6.02967190296508779603759044043, 7.22346333189074729301208498018, 8.011556403336425792757951435732, 8.336087291148535323846459624034
