L(s) = 1 | + (1.34 + 0.437i)2-s + (0.809 − 0.587i)3-s + (−1.61 − 1.17i)4-s + (−2.16 − 6.65i)5-s + (1.34 − 0.437i)6-s + (4.15 − 5.72i)7-s + (−4.98 − 6.86i)8-s + (−2.47 + 7.60i)9-s − 9.89i·10-s − 1.99·12-s + (16.1 + 5.24i)13-s + (8.09 − 5.87i)14-s + (−5.66 − 4.11i)15-s + (−1.23 − 3.80i)16-s + (4.03 − 1.31i)17-s + (−6.65 + 9.15i)18-s + ⋯ |
L(s) = 1 | + (0.672 + 0.218i)2-s + (0.269 − 0.195i)3-s + (−0.404 − 0.293i)4-s + (−0.432 − 1.33i)5-s + (0.224 − 0.0728i)6-s + (0.593 − 0.817i)7-s + (−0.623 − 0.858i)8-s + (−0.274 + 0.845i)9-s − 0.989i·10-s − 0.166·12-s + (1.24 + 0.403i)13-s + (0.577 − 0.419i)14-s + (−0.377 − 0.274i)15-s + (−0.0772 − 0.237i)16-s + (0.237 − 0.0771i)17-s + (−0.369 + 0.508i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.42402 - 1.00823i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42402 - 1.00823i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
good | 2 | \( 1 + (-1.34 - 0.437i)T + (3.23 + 2.35i)T^{2} \) |
| 3 | \( 1 + (-0.809 + 0.587i)T + (2.78 - 8.55i)T^{2} \) |
| 5 | \( 1 + (2.16 + 6.65i)T + (-20.2 + 14.6i)T^{2} \) |
| 7 | \( 1 + (-4.15 + 5.72i)T + (-15.1 - 46.6i)T^{2} \) |
| 13 | \( 1 + (-16.1 - 5.24i)T + (136. + 99.3i)T^{2} \) |
| 17 | \( 1 + (-4.03 + 1.31i)T + (233. - 169. i)T^{2} \) |
| 19 | \( 1 + (-9.97 - 13.7i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + 9T + 529T^{2} \) |
| 29 | \( 1 + (-13.3 + 18.3i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (-15.1 + 46.6i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (13.7 + 9.99i)T + (423. + 1.30e3i)T^{2} \) |
| 41 | \( 1 + (-9.97 - 13.7i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 - 46.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (25.8 - 18.8i)T + (682. - 2.10e3i)T^{2} \) |
| 53 | \( 1 + (-4.94 + 15.2i)T + (-2.27e3 - 1.65e3i)T^{2} \) |
| 59 | \( 1 + (-57.4 - 41.7i)T + (1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (10.7 - 3.49i)T + (3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + 31T + 4.48e3T^{2} \) |
| 71 | \( 1 + (22.5 + 69.4i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (23.2 - 32.0i)T + (-1.64e3 - 5.06e3i)T^{2} \) |
| 79 | \( 1 + (-149. - 48.5i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-33.6 + 10.9i)T + (5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 + 9T + 7.92e3T^{2} \) |
| 97 | \( 1 + (5.25 - 16.1i)T + (-7.61e3 - 5.53e3i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.45674042952899291601173845987, −12.25971634386568914705893670310, −11.12945805638613859496544847965, −9.741387740342870666596307349867, −8.512082943909344053873025350889, −7.79166443771412766814848612105, −5.96377448350234449641293224842, −4.78230135761053506258359653999, −3.96062567842349073889481108473, −1.13783356645452998626987654900,
2.91403080181752987246630565411, 3.68863002622870899429040192593, 5.34499349751824654728736190708, 6.65013349165572368973319721142, 8.221251123096217313856624356972, 8.983278940399667528882423005023, 10.57011419246957849622268239529, 11.59356609180422923305340005634, 12.22956789777711602011977549355, 13.63240748196720249625600808445