| L(s) = 1 | + 2-s − 3.14·3-s + 4-s + 1.53·5-s − 3.14·6-s − 4.82·7-s + 8-s + 6.91·9-s + 1.53·10-s − 1.09·11-s − 3.14·12-s + 1.76·13-s − 4.82·14-s − 4.82·15-s + 16-s + 2.94·17-s + 6.91·18-s − 0.825·19-s + 1.53·20-s + 15.1·21-s − 1.09·22-s − 0.243·23-s − 3.14·24-s − 2.65·25-s + 1.76·26-s − 12.3·27-s − 4.82·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.81·3-s + 0.5·4-s + 0.685·5-s − 1.28·6-s − 1.82·7-s + 0.353·8-s + 2.30·9-s + 0.484·10-s − 0.329·11-s − 0.909·12-s + 0.490·13-s − 1.28·14-s − 1.24·15-s + 0.250·16-s + 0.714·17-s + 1.63·18-s − 0.189·19-s + 0.342·20-s + 3.31·21-s − 0.233·22-s − 0.0507·23-s − 0.642·24-s − 0.530·25-s + 0.347·26-s − 2.37·27-s − 0.911·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.171713785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.171713785\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 + 3.14T + 3T^{2} \) |
| 5 | \( 1 - 1.53T + 5T^{2} \) |
| 7 | \( 1 + 4.82T + 7T^{2} \) |
| 11 | \( 1 + 1.09T + 11T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 17 | \( 1 - 2.94T + 17T^{2} \) |
| 19 | \( 1 + 0.825T + 19T^{2} \) |
| 23 | \( 1 + 0.243T + 23T^{2} \) |
| 29 | \( 1 + 5.57T + 29T^{2} \) |
| 31 | \( 1 + 5.73T + 31T^{2} \) |
| 41 | \( 1 - 2.91T + 41T^{2} \) |
| 43 | \( 1 - 4.37T + 43T^{2} \) |
| 47 | \( 1 - 2.26T + 47T^{2} \) |
| 53 | \( 1 - 6.14T + 53T^{2} \) |
| 59 | \( 1 - 8.82T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 - 8.98T + 67T^{2} \) |
| 71 | \( 1 + 1.21T + 71T^{2} \) |
| 73 | \( 1 - 8.79T + 73T^{2} \) |
| 79 | \( 1 - 2.15T + 79T^{2} \) |
| 83 | \( 1 - 5.09T + 83T^{2} \) |
| 89 | \( 1 - 8.97T + 89T^{2} \) |
| 97 | \( 1 - 4.35T + 97T^{2} \) |
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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
−9.221984253116703782379061830396, −7.60062485410483226286402050693, −6.89232659647045190801631455250, −6.22115508057097223058903531295, −5.75741363307623968794715923483, −5.33931654598018118937290578042, −4.11762592653758361432272983653, −3.41318501251781655326701362352, −2.07074291301131409229417896247, −0.64700941864325567050253832360,
0.64700941864325567050253832360, 2.07074291301131409229417896247, 3.41318501251781655326701362352, 4.11762592653758361432272983653, 5.33931654598018118937290578042, 5.75741363307623968794715923483, 6.22115508057097223058903531295, 6.89232659647045190801631455250, 7.60062485410483226286402050693, 9.221984253116703782379061830396
