| L(s) = 1 | + i·3-s − 1.62·5-s + 0.682·7-s − 9-s + 1.65i·11-s + 6.88·13-s − 1.62i·15-s − 5.88i·17-s − 5.78·19-s + 0.682i·21-s + 4.15i·23-s − 2.37·25-s − i·27-s + 5.64·29-s − 9.48i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.724·5-s + 0.258·7-s − 0.333·9-s + 0.497i·11-s + 1.90·13-s − 0.418i·15-s − 1.42i·17-s − 1.32·19-s + 0.148i·21-s + 0.865i·23-s − 0.474·25-s − 0.192i·27-s + 1.04·29-s − 1.70i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.136i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.582849970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.582849970\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 37 | \( 1 + (0.243 + 6.07i)T \) |
| good | 5 | \( 1 + 1.62T + 5T^{2} \) |
| 7 | \( 1 - 0.682T + 7T^{2} \) |
| 11 | \( 1 - 1.65iT - 11T^{2} \) |
| 13 | \( 1 - 6.88T + 13T^{2} \) |
| 17 | \( 1 + 5.88iT - 17T^{2} \) |
| 19 | \( 1 + 5.78T + 19T^{2} \) |
| 23 | \( 1 - 4.15iT - 23T^{2} \) |
| 29 | \( 1 - 5.64T + 29T^{2} \) |
| 31 | \( 1 + 9.48iT - 31T^{2} \) |
| 41 | \( 1 - 2.20T + 41T^{2} \) |
| 43 | \( 1 - 5.62T + 43T^{2} \) |
| 47 | \( 1 - 2.88T + 47T^{2} \) |
| 53 | \( 1 - 4.56iT - 53T^{2} \) |
| 59 | \( 1 - 6.60T + 59T^{2} \) |
| 61 | \( 1 - 0.499T + 61T^{2} \) |
| 67 | \( 1 + 10.9iT - 67T^{2} \) |
| 71 | \( 1 - 1.34T + 71T^{2} \) |
| 73 | \( 1 + 9.65T + 73T^{2} \) |
| 79 | \( 1 + 3.25iT - 79T^{2} \) |
| 83 | \( 1 - 12.4iT - 83T^{2} \) |
| 89 | \( 1 - 10.7iT - 89T^{2} \) |
| 97 | \( 1 + 11.0iT - 97T^{2} \) |
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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
−8.522637098541916319267356277621, −7.917005166431681794411743165454, −7.17267969625934424694556294792, −6.20255995775254660237246342654, −5.57077719057687444008922474549, −4.40794326188359192812315374948, −4.10105964322228178761958525461, −3.16952102619977215728420223603, −2.04669370599155996996238986903, −0.61673806124105840478540166070,
0.927219058515872734680013679116, 1.85961294717125087889486876872, 3.14130865815599589951318761796, 3.89203526334521603448623746400, 4.60267165969138627230306820634, 5.91738465181263879043513606093, 6.28153960613233147190169922053, 7.02420041875821353033018822563, 8.203537140043693705588491351333, 8.405273282080504895260765636622
