| L(s) = 1 | + 9.71i·7-s + 1.41i·11-s − 22.4i·13-s − 13.7·17-s − 11.7·19-s + 31.7·23-s − 16.5i·29-s − 7.15·31-s − 45.4i·37-s − 11.5i·41-s − 14.2i·43-s + 77.9·47-s − 45.4·49-s − 11.5·53-s − 47.6i·59-s + ⋯ |
| L(s) = 1 | + 1.38i·7-s + 0.128i·11-s − 1.72i·13-s − 0.808·17-s − 0.616·19-s + 1.37·23-s − 0.571i·29-s − 0.230·31-s − 1.22i·37-s − 0.281i·41-s − 0.332i·43-s + 1.65·47-s − 0.927·49-s − 0.217·53-s − 0.808i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.724750306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.724750306\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 9.71iT - 49T^{2} \) |
| 11 | \( 1 - 1.41iT - 121T^{2} \) |
| 13 | \( 1 + 22.4iT - 169T^{2} \) |
| 17 | \( 1 + 13.7T + 289T^{2} \) |
| 19 | \( 1 + 11.7T + 361T^{2} \) |
| 23 | \( 1 - 31.7T + 529T^{2} \) |
| 29 | \( 1 + 16.5iT - 841T^{2} \) |
| 31 | \( 1 + 7.15T + 961T^{2} \) |
| 37 | \( 1 + 45.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 11.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 14.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 77.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 11.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 47.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 91.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 35.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 140. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 16.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 61.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + 43.2T + 6.88e3T^{2} \) |
| 89 | \( 1 - 151. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 76.1iT - 9.40e3T^{2} \) |
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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
−8.895017221656729446533484503107, −8.429583159021229937679725282082, −7.50913871899205037189503499083, −6.58436489479135723382876617695, −5.60613393993938879115457793689, −5.24369888625091856907282321240, −3.99885726802391289474734797309, −2.83852907324357829192662836837, −2.20054498943706041478817208230, −0.56567716080644422086428499007,
0.908983262586366877216868089955, 2.01975811337507519010827595026, 3.34112308167518234234016637196, 4.31992149173779345919648387987, 4.74233624090926193763927069210, 6.15629959906258747557468784444, 6.96042270091832185709533772260, 7.26925340255283833911429485165, 8.516261830855068371486043190790, 9.080826650279056165899197148395
