L(s) = 1 | + 1.08·3-s − 2.61i·5-s + (2.61 − 0.414i)7-s − 1.82·9-s + 2i·11-s − 4.77i·13-s − 2.82i·15-s + 3.06i·17-s + 4.14·19-s + (2.82 − 0.448i)21-s + 7.65i·23-s − 1.82·25-s − 5.22·27-s + 3.65·29-s − 3.06·31-s + ⋯ |
L(s) = 1 | + 0.624·3-s − 1.16i·5-s + (0.987 − 0.156i)7-s − 0.609·9-s + 0.603i·11-s − 1.32i·13-s − 0.730i·15-s + 0.742i·17-s + 0.950·19-s + (0.617 − 0.0978i)21-s + 1.59i·23-s − 0.365·25-s − 1.00·27-s + 0.679·29-s − 0.549·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 + 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.44110 - 0.468142i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.44110 - 0.468142i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.61 + 0.414i)T \) |
good | 3 | \( 1 - 1.08T + 3T^{2} \) |
| 5 | \( 1 + 2.61iT - 5T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 + 4.77iT - 13T^{2} \) |
| 17 | \( 1 - 3.06iT - 17T^{2} \) |
| 19 | \( 1 - 4.14T + 19T^{2} \) |
| 23 | \( 1 - 7.65iT - 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 + 3.06T + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 - 9.55iT - 41T^{2} \) |
| 43 | \( 1 + 3.65iT - 43T^{2} \) |
| 47 | \( 1 + 7.39T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 8.47T + 59T^{2} \) |
| 61 | \( 1 - 2.61iT - 61T^{2} \) |
| 67 | \( 1 - 15.6iT - 67T^{2} \) |
| 71 | \( 1 + 8.82iT - 71T^{2} \) |
| 73 | \( 1 + 12.6iT - 73T^{2} \) |
| 79 | \( 1 - 12.8iT - 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 + 2.16iT - 89T^{2} \) |
| 97 | \( 1 + 13.5iT - 97T^{2} \) |
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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
−12.22341711598299224374571383114, −11.31108169557033368732710940248, −10.09109343964340354348162741885, −9.028234743238274669939263869075, −8.204120196178458737340759376553, −7.58216102595077472034077049256, −5.61940615199338735197907410939, −4.86155165324298046699039662024, −3.34758402955247178487605076665, −1.52214246454495651836127650903,
2.26758508862851611296465021458, 3.38136351588905753384185367880, 4.94196251110733237493944550748, 6.37651356493410621916977935445, 7.37468695279959692471631846616, 8.456235093140956031078482065266, 9.243679587663804398895403718479, 10.61840693672538254358067933400, 11.32923141387762628809203427686, 12.07437939600384334077417069062