L(s) = 1 | + (−1.99 − 0.143i)2-s + 1.73i·3-s + (3.95 + 0.572i)4-s + 8.96i·5-s + (0.248 − 3.45i)6-s − 7.15·7-s + (−7.81 − 1.71i)8-s − 2.99·9-s + (1.28 − 17.8i)10-s + 9.63·11-s + (−0.991 + 6.85i)12-s + (0.746 − 12.9i)13-s + (14.2 + 1.02i)14-s − 15.5·15-s + (15.3 + 4.53i)16-s − 18.0·17-s + ⋯ |
L(s) = 1 | + (−0.997 − 0.0717i)2-s + 0.577i·3-s + (0.989 + 0.143i)4-s + 1.79i·5-s + (0.0414 − 0.575i)6-s − 1.02·7-s + (−0.976 − 0.213i)8-s − 0.333·9-s + (0.128 − 1.78i)10-s + 0.876·11-s + (−0.0826 + 0.571i)12-s + (0.0574 − 0.998i)13-s + (1.01 + 0.0733i)14-s − 1.03·15-s + (0.959 + 0.283i)16-s − 1.06·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.199i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.979 - 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0529091 + 0.524338i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0529091 + 0.524338i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.99 + 0.143i)T \) |
| 3 | \( 1 - 1.73iT \) |
| 13 | \( 1 + (-0.746 + 12.9i)T \) |
good | 5 | \( 1 - 8.96iT - 25T^{2} \) |
| 7 | \( 1 + 7.15T + 49T^{2} \) |
| 11 | \( 1 - 9.63T + 121T^{2} \) |
| 17 | \( 1 + 18.0T + 289T^{2} \) |
| 19 | \( 1 + 23.1T + 361T^{2} \) |
| 23 | \( 1 - 25.4iT - 529T^{2} \) |
| 29 | \( 1 + 2.34T + 841T^{2} \) |
| 31 | \( 1 - 40.4T + 961T^{2} \) |
| 37 | \( 1 - 20.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 43.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 20.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 19.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 55.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 45.1T + 3.48e3T^{2} \) |
| 61 | \( 1 - 106.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 62.9T + 4.48e3T^{2} \) |
| 71 | \( 1 - 4.89T + 5.04e3T^{2} \) |
| 73 | \( 1 + 70.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 86.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 68.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 64.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 118. iT - 9.40e3T^{2} \) |
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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
−13.11968206987292573042275010676, −11.65189104607436014931204844186, −10.88772065935331887463877111593, −10.12883665703414142121072331179, −9.403259531649217506893909762066, −8.039265837608781248847700588602, −6.67023150462633251091607024947, −6.27548350637670257942674167636, −3.61582981237980548187962956345, −2.65158983865884880054531626172,
0.43375792475421206358538970675, 1.98073463191256716660789161212, 4.31328159364342156419331675066, 6.12466301831082297912786772487, 6.87925037910215504857082370117, 8.553737507552991724987597115381, 8.847751291869425126896681467170, 9.825640415044889305294050707046, 11.32592923257334022797199293435, 12.34625986987376894050504628220