| L(s) = 1 | + 4.38·5-s + 7-s + 5.38·11-s − 2.54·13-s − 2.58·17-s + 6.72·19-s + 0.800·23-s + 14.1·25-s + 3.74·29-s − 3.39·31-s + 4.38·35-s + 4.38·37-s − 6.39·41-s + 0.763·43-s + 8.26·47-s + 49-s − 4.94·53-s + 23.5·55-s − 5.57·59-s − 8.28·61-s − 11.1·65-s + 1.89·67-s − 1.34·71-s − 8.65·73-s + 5.38·77-s + 13.2·79-s − 3.72·83-s + ⋯ |
| L(s) = 1 | + 1.95·5-s + 0.377·7-s + 1.62·11-s − 0.705·13-s − 0.625·17-s + 1.54·19-s + 0.166·23-s + 2.83·25-s + 0.695·29-s − 0.609·31-s + 0.740·35-s + 0.720·37-s − 0.998·41-s + 0.116·43-s + 1.20·47-s + 0.142·49-s − 0.679·53-s + 3.17·55-s − 0.725·59-s − 1.06·61-s − 1.38·65-s + 0.231·67-s − 0.159·71-s − 1.01·73-s + 0.613·77-s + 1.49·79-s − 0.408·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.015019360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.015019360\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 4.38T + 5T^{2} \) |
| 11 | \( 1 - 5.38T + 11T^{2} \) |
| 13 | \( 1 + 2.54T + 13T^{2} \) |
| 17 | \( 1 + 2.58T + 17T^{2} \) |
| 19 | \( 1 - 6.72T + 19T^{2} \) |
| 23 | \( 1 - 0.800T + 23T^{2} \) |
| 29 | \( 1 - 3.74T + 29T^{2} \) |
| 31 | \( 1 + 3.39T + 31T^{2} \) |
| 37 | \( 1 - 4.38T + 37T^{2} \) |
| 41 | \( 1 + 6.39T + 41T^{2} \) |
| 43 | \( 1 - 0.763T + 43T^{2} \) |
| 47 | \( 1 - 8.26T + 47T^{2} \) |
| 53 | \( 1 + 4.94T + 53T^{2} \) |
| 59 | \( 1 + 5.57T + 59T^{2} \) |
| 61 | \( 1 + 8.28T + 61T^{2} \) |
| 67 | \( 1 - 1.89T + 67T^{2} \) |
| 71 | \( 1 + 1.34T + 71T^{2} \) |
| 73 | \( 1 + 8.65T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 + 3.72T + 83T^{2} \) |
| 89 | \( 1 + 6.99T + 89T^{2} \) |
| 97 | \( 1 - 2.96T + 97T^{2} \) |
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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
−7.56566462232704814237621251120, −6.89787247870565617141759322499, −6.33554311931507512237003427839, −5.72958362218120559683216699304, −5.04234221229493937163527486949, −4.43030818379174402772418358871, −3.29265262190363711714625792426, −2.49433738744905614253484444343, −1.66373333869864534560562723323, −1.08455478689994092673817065389,
1.08455478689994092673817065389, 1.66373333869864534560562723323, 2.49433738744905614253484444343, 3.29265262190363711714625792426, 4.43030818379174402772418358871, 5.04234221229493937163527486949, 5.72958362218120559683216699304, 6.33554311931507512237003427839, 6.89787247870565617141759322499, 7.56566462232704814237621251120
