| L(s) = 1 | + 1.41·2-s + 2.00·4-s − 2.64i·7-s + 2.82·8-s + 11.2i·11-s − 22.9i·13-s − 3.74i·14-s + 4.00·16-s − 8.36·17-s − 3.57·19-s + 15.9i·22-s + 3.92·23-s − 32.4i·26-s − 5.29i·28-s + 43.5i·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s − 0.377i·7-s + 0.353·8-s + 1.02i·11-s − 1.76i·13-s − 0.267i·14-s + 0.250·16-s − 0.491·17-s − 0.187·19-s + 0.724i·22-s + 0.170·23-s − 1.24i·26-s − 0.188i·28-s + 1.50i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.828244421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.828244421\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 11 | \( 1 - 11.2iT - 121T^{2} \) |
| 13 | \( 1 + 22.9iT - 169T^{2} \) |
| 17 | \( 1 + 8.36T + 289T^{2} \) |
| 19 | \( 1 + 3.57T + 361T^{2} \) |
| 23 | \( 1 - 3.92T + 529T^{2} \) |
| 29 | \( 1 - 43.5iT - 841T^{2} \) |
| 31 | \( 1 - 46.3T + 961T^{2} \) |
| 37 | \( 1 + 27.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 32.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 65.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 5.65T + 2.20e3T^{2} \) |
| 53 | \( 1 + 24.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 58.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 28.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 10.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 20.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 113. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 133.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 30.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 79.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 26.1iT - 9.40e3T^{2} \) |
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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
−8.217110980369516333668772013613, −7.39188451898394250009976911229, −6.90803864301228306589956853999, −5.94639482415769504082384363361, −5.17362320613361361882878616535, −4.54515634383942815976492690118, −3.59294884161235675990559486664, −2.80739515175390696514578022001, −1.78498532317597182303929452392, −0.48999737194712312911948942936,
1.16742240616103894926799860880, 2.31382387820415026350019041416, 3.05193524227502203880312875563, 4.23309417443300687199745843722, 4.57907979301328355014516950798, 5.74871534981583856116352702777, 6.35870150762991365253627378444, 6.86414150242303187459115083188, 8.016237250331358366566619881078, 8.563855745833544742773854279036
