L(s) = 1 | − 3.54·5-s − 5.04i·7-s − 1.25i·11-s − 4.25i·13-s + 0.236i·17-s − 0.840i·19-s + 7.74·23-s + 7.58·25-s − 10.3i·29-s − 4.69·31-s + 17.8i·35-s + 5.58·37-s + (−6.33 − 0.944i)41-s + 7.82·43-s − 2.07i·47-s + ⋯ |
L(s) = 1 | − 1.58·5-s − 1.90i·7-s − 0.378i·11-s − 1.18i·13-s + 0.0574i·17-s − 0.192i·19-s + 1.61·23-s + 1.51·25-s − 1.92i·29-s − 0.843·31-s + 3.02i·35-s + 0.918·37-s + (−0.989 − 0.147i)41-s + 1.19·43-s − 0.302i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.147i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7832827729\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7832827729\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 41 | \( 1 + (6.33 + 0.944i)T \) |
good | 5 | \( 1 + 3.54T + 5T^{2} \) |
| 7 | \( 1 + 5.04iT - 7T^{2} \) |
| 11 | \( 1 + 1.25iT - 11T^{2} \) |
| 13 | \( 1 + 4.25iT - 13T^{2} \) |
| 17 | \( 1 - 0.236iT - 17T^{2} \) |
| 19 | \( 1 + 0.840iT - 19T^{2} \) |
| 23 | \( 1 - 7.74T + 23T^{2} \) |
| 29 | \( 1 + 10.3iT - 29T^{2} \) |
| 31 | \( 1 + 4.69T + 31T^{2} \) |
| 37 | \( 1 - 5.58T + 37T^{2} \) |
| 43 | \( 1 - 7.82T + 43T^{2} \) |
| 47 | \( 1 + 2.07iT - 47T^{2} \) |
| 53 | \( 1 + 1.29iT - 53T^{2} \) |
| 59 | \( 1 + 7.40T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 - 5.18iT - 67T^{2} \) |
| 71 | \( 1 - 6.74iT - 71T^{2} \) |
| 73 | \( 1 - 8.41T + 73T^{2} \) |
| 79 | \( 1 + 11.1iT - 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 - 3.21iT - 89T^{2} \) |
| 97 | \( 1 + 3.94iT - 97T^{2} \) |
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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
−8.035419751144841258368257386641, −7.61745900996139723571045893155, −7.18042591755877615027149055106, −6.22088569069694647646582528683, −5.00318824848991520541541287539, −4.25237080196311322053829734978, −3.66321955009736361882578033806, −2.94308105249600127255203532016, −0.975034342246710721281944473087, −0.31828476049705774950916602649,
1.58904750136777301845811190450, 2.77192289579614970988582257170, 3.48708936204660045692403064359, 4.59369553207932897853619166278, 5.10389221330573552411973038372, 6.15085246549233832814364798173, 7.01274085665331564289004322119, 7.60996813396534823435747386005, 8.580028226071099955562588225633, 8.964237543070500819800655551089