| L(s) = 1 | + (−2.72 − 1.26i)3-s + (−0.689 + 4.95i)5-s − 0.735i·7-s + (5.82 + 6.86i)9-s − 10.9i·11-s − 21.1i·13-s + (8.11 − 12.6i)15-s − 7.03·17-s − 23.1·19-s + (−0.927 + 2.00i)21-s − 24.7·23-s + (−24.0 − 6.82i)25-s + (−7.21 − 26.0i)27-s − 32.3i·29-s + 34.9·31-s + ⋯ |
| L(s) = 1 | + (−0.907 − 0.420i)3-s + (−0.137 + 0.990i)5-s − 0.105i·7-s + (0.647 + 0.762i)9-s − 0.995i·11-s − 1.63i·13-s + (0.541 − 0.840i)15-s − 0.413·17-s − 1.21·19-s + (−0.0441 + 0.0953i)21-s − 1.07·23-s + (−0.962 − 0.272i)25-s + (−0.267 − 0.963i)27-s − 1.11i·29-s + 1.12·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.541 + 0.840i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.541 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.279947 - 0.512987i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.279947 - 0.512987i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.72 + 1.26i)T \) |
| 5 | \( 1 + (0.689 - 4.95i)T \) |
| good | 7 | \( 1 + 0.735iT - 49T^{2} \) |
| 11 | \( 1 + 10.9iT - 121T^{2} \) |
| 13 | \( 1 + 21.1iT - 169T^{2} \) |
| 17 | \( 1 + 7.03T + 289T^{2} \) |
| 19 | \( 1 + 23.1T + 361T^{2} \) |
| 23 | \( 1 + 24.7T + 529T^{2} \) |
| 29 | \( 1 + 32.3iT - 841T^{2} \) |
| 31 | \( 1 - 34.9T + 961T^{2} \) |
| 37 | \( 1 + 37.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 39.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 22.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 39.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 60.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 7.79iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 11.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 33.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 96.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 134. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 121.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 90.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 53.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 115. iT - 9.40e3T^{2} \) |
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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
−11.44127694068565107479016454341, −10.68387717761609215626701490395, −10.13906903101663305075342110310, −8.333838305590120229492029915052, −7.52405642002791425299283162094, −6.30221189908955298440743776005, −5.73500903592233743477642005048, −4.07496334669074033529700467753, −2.52660127541498326016691504960, −0.34285760598174426797576369275,
1.70886955074925118310841616316, 4.26608441468320276497056817350, 4.66705342601054466290572088158, 6.09163876736711514637891654026, 7.02576196381714374124277524675, 8.530183334357071432734297520044, 9.410461883244732884124003140507, 10.27015273488240181560640990603, 11.47724190951364477722410529501, 12.13554300568798031519158291897
