L(s) = 1 | + (4.22 − 3.75i)2-s − 25.0i·3-s + (3.78 − 31.7i)4-s + 25i·5-s + (−94.1 − 105. i)6-s + 103.·7-s + (−103. − 148. i)8-s − 384.·9-s + (93.9 + 105. i)10-s + 740. i·11-s + (−796. − 94.7i)12-s − 892. i·13-s + (438. − 389. i)14-s + 626.·15-s + (−995. − 240. i)16-s + 1.13e3·17-s + ⋯ |
L(s) = 1 | + (0.747 − 0.664i)2-s − 1.60i·3-s + (0.118 − 0.992i)4-s + 0.447i·5-s + (−1.06 − 1.20i)6-s + 0.799·7-s + (−0.571 − 0.820i)8-s − 1.58·9-s + (0.296 + 0.334i)10-s + 1.84i·11-s + (−1.59 − 0.189i)12-s − 1.46i·13-s + (0.597 − 0.530i)14-s + 0.718·15-s + (−0.972 − 0.234i)16-s + 0.955·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.710155 - 2.26452i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.710155 - 2.26452i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-4.22 + 3.75i)T \) |
| 5 | \( 1 - 25iT \) |
good | 3 | \( 1 + 25.0iT - 243T^{2} \) |
| 7 | \( 1 - 103.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 740. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 892. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.13e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.15e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 1.60e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.15e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 4.95e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.40e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 3.78e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.30e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 8.00e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.43e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.20e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 2.82e3iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 5.49e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.28e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.08e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.02e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.39e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.17e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.41e4T + 8.58e9T^{2} \) |
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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
−14.47119336976575317226560504683, −13.15599282552607515430311017225, −12.48411892210895176470009465203, −11.46784092116231564225077832174, −10.07011024416590438853636665883, −7.84501949273412000323688316077, −6.76017320833967476675872180965, −5.11799189552118532592052781462, −2.64642627274454206792165093060, −1.23457771820016463361882648431,
3.54955716641439625793285731279, 4.72358126466787606899155806376, 5.87965806317797403681386465845, 8.190009431663367257003541297932, 9.181397888428049285223917762225, 10.95785166957701712141949821192, 11.87656439932100976283731297774, 13.85584080711270808611455143052, 14.45340836307177643009967417113, 15.70218472151835090392134709066