| L(s) = 1 | − 1.41·2-s + (−2.82 + i)3-s + 2.00·4-s + (4.00 − 1.41i)6-s − 7i·7-s − 2.82·8-s + (7.00 − 5.65i)9-s + 8.48i·11-s + (−5.65 + 2.00i)12-s + 25i·13-s + 9.89i·14-s + 4.00·16-s + 25.4·17-s + (−9.89 + 8.00i)18-s + 7·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + (−0.942 + 0.333i)3-s + 0.500·4-s + (0.666 − 0.235i)6-s − i·7-s − 0.353·8-s + (0.777 − 0.628i)9-s + 0.771i·11-s + (−0.471 + 0.166i)12-s + 1.92i·13-s + 0.707i·14-s + 0.250·16-s + 1.49·17-s + (−0.549 + 0.444i)18-s + 0.368·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.703812 + 0.299013i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.703812 + 0.299013i\) |
| \(L(2)\) |
|
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 | \( 1 + (2.82 - i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 7iT - 49T^{2} \) |
| 11 | \( 1 - 8.48iT - 121T^{2} \) |
| 13 | \( 1 - 25iT - 169T^{2} \) |
| 17 | \( 1 - 25.4T + 289T^{2} \) |
| 19 | \( 1 - 7T + 361T^{2} \) |
| 23 | \( 1 - 25.4T + 529T^{2} \) |
| 29 | \( 1 - 42.4iT - 841T^{2} \) |
| 31 | \( 1 + 7T + 961T^{2} \) |
| 37 | \( 1 - 2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 8.48iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 41iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 - 59.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 33.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + T + 3.72e3T^{2} \) |
| 67 | \( 1 - 17iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 42.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 70iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 58T + 6.24e3T^{2} \) |
| 83 | \( 1 + 118.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 135. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 49iT - 9.40e3T^{2} \) |
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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
−12.53885176285912752435020006039, −11.71556558964148467290917202942, −10.77583297310962356609096521028, −9.940291083765254564322826354032, −9.086040166258983633289739969126, −7.29584264248064017982330285295, −6.81439803170785634531645338046, −5.18757978943452404383970399269, −3.87808703953683476607499867658, −1.29662393165160877032271142503,
0.839885226531149931863732304929, 2.94996113755282503885502422581, 5.42135894823287621031263127977, 5.94436517204948848102124945715, 7.51454640340984204066524201763, 8.334812015112123815353315350416, 9.738288792759138332762344006542, 10.63727418589806541321300690490, 11.60833715248973216112603424379, 12.40869074315240279147159524378
