| L(s) = 1 | − 2-s + 1.52·3-s + 4-s − 3.19·5-s − 1.52·6-s − 7-s − 8-s − 0.686·9-s + 3.19·10-s − 1.28·11-s + 1.52·12-s + 1.34·13-s + 14-s − 4.85·15-s + 16-s + 5.77·17-s + 0.686·18-s − 7.04·19-s − 3.19·20-s − 1.52·21-s + 1.28·22-s − 5.21·23-s − 1.52·24-s + 5.18·25-s − 1.34·26-s − 5.60·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.878·3-s + 0.5·4-s − 1.42·5-s − 0.620·6-s − 0.377·7-s − 0.353·8-s − 0.228·9-s + 1.00·10-s − 0.388·11-s + 0.439·12-s + 0.374·13-s + 0.267·14-s − 1.25·15-s + 0.250·16-s + 1.39·17-s + 0.161·18-s − 1.61·19-s − 0.713·20-s − 0.331·21-s + 0.274·22-s − 1.08·23-s − 0.310·24-s + 1.03·25-s − 0.264·26-s − 1.07·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7428028480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7428028480\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 - 1.52T + 3T^{2} \) |
| 5 | \( 1 + 3.19T + 5T^{2} \) |
| 11 | \( 1 + 1.28T + 11T^{2} \) |
| 13 | \( 1 - 1.34T + 13T^{2} \) |
| 17 | \( 1 - 5.77T + 17T^{2} \) |
| 19 | \( 1 + 7.04T + 19T^{2} \) |
| 23 | \( 1 + 5.21T + 23T^{2} \) |
| 29 | \( 1 + 2.06T + 29T^{2} \) |
| 31 | \( 1 + 1.05T + 31T^{2} \) |
| 37 | \( 1 - 0.620T + 37T^{2} \) |
| 41 | \( 1 + 6.09T + 41T^{2} \) |
| 43 | \( 1 + 9.64T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 2.46T + 59T^{2} \) |
| 61 | \( 1 + 9.99T + 61T^{2} \) |
| 67 | \( 1 - 15.7T + 67T^{2} \) |
| 71 | \( 1 - 7.43T + 71T^{2} \) |
| 73 | \( 1 - 5.84T + 73T^{2} \) |
| 79 | \( 1 + 2.91T + 79T^{2} \) |
| 83 | \( 1 + 2.31T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 - 9.65T + 97T^{2} \) |
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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
−8.101576164094373202893222320589, −7.74591669328070811741625810185, −6.94673300167821749676390521122, −6.12082864329018874716380173784, −5.25978927758045613568042549269, −3.94218180302843175295268483498, −3.67694470272822944856569545762, −2.79823600349381824924250261167, −1.91507916395981168773304031945, −0.45877292100159491106272449041,
0.45877292100159491106272449041, 1.91507916395981168773304031945, 2.79823600349381824924250261167, 3.67694470272822944856569545762, 3.94218180302843175295268483498, 5.25978927758045613568042549269, 6.12082864329018874716380173784, 6.94673300167821749676390521122, 7.74591669328070811741625810185, 8.101576164094373202893222320589
