L(s) = 1 | + (1.56 + 2.55i)3-s − 2.23i·5-s − 0.823·7-s + (−4.07 + 8.02i)9-s + 2.87i·11-s − 14.8·13-s + (5.71 − 3.50i)15-s + 8.06i·17-s − 17.2·19-s + (−1.29 − 2.10i)21-s − 42.0i·23-s − 5.00·25-s + (−26.9 + 2.16i)27-s + 6.39i·29-s + 7.47·31-s + ⋯ |
L(s) = 1 | + (0.522 + 0.852i)3-s − 0.447i·5-s − 0.117·7-s + (−0.453 + 0.891i)9-s + 0.261i·11-s − 1.14·13-s + (0.381 − 0.233i)15-s + 0.474i·17-s − 0.908·19-s + (−0.0615 − 0.100i)21-s − 1.82i·23-s − 0.200·25-s + (−0.996 + 0.0800i)27-s + 0.220i·29-s + 0.241·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 + 0.522i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1118025767\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1118025767\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.56 - 2.55i)T \) |
| 5 | \( 1 + 2.23iT \) |
good | 7 | \( 1 + 0.823T + 49T^{2} \) |
| 11 | \( 1 - 2.87iT - 121T^{2} \) |
| 13 | \( 1 + 14.8T + 169T^{2} \) |
| 17 | \( 1 - 8.06iT - 289T^{2} \) |
| 19 | \( 1 + 17.2T + 361T^{2} \) |
| 23 | \( 1 + 42.0iT - 529T^{2} \) |
| 29 | \( 1 - 6.39iT - 841T^{2} \) |
| 31 | \( 1 - 7.47T + 961T^{2} \) |
| 37 | \( 1 + 18.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 16.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 25.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 18.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 68.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 68.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 61.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 120.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 69.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 143.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 124.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 49.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 49.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 78.7T + 9.40e3T^{2} \) |
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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
−10.33562587346716835171424199064, −9.455992835691397128110407936492, −8.718899523581328629488477953766, −8.070062260232963266745570874530, −7.01921704343105768660668288550, −5.90825118677099211228416742674, −4.74858667367134017495230672063, −4.33519465116604546239106985662, −3.01763751685647172984222320594, −2.01726384003451675551176958370,
0.03011477526689296992353660223, 1.68733969346747506514252285819, 2.73774177857528972501431879096, 3.61655366254432137133880273609, 5.00421496856089401752853916630, 6.10535549661835643198779418031, 6.92006727548065349371641064766, 7.60111978223379039118821503762, 8.352890441853979446748825951396, 9.392916493201148398867438709081