L(s) = 1 | + (0.694 − 3.93i)2-s + (3.82 + 21.6i)3-s + (−15.0 − 5.47i)4-s + (−48.1 + 40.4i)5-s + 88.0·6-s + (134. − 112. i)7-s + (−32 + 55.4i)8-s + (−226. + 82.4i)9-s + (125. + 217. i)10-s + (−241. + 418. i)11-s + (61.1 − 346. i)12-s + (−625. − 227. i)13-s + (−350. − 607. i)14-s + (−1.06e3 − 889. i)15-s + (196. + 164. i)16-s + (−2.16e3 + 789. i)17-s + ⋯ |
L(s) = 1 | + (0.122 − 0.696i)2-s + (0.245 + 1.38i)3-s + (−0.469 − 0.171i)4-s + (−0.861 + 0.723i)5-s + 0.998·6-s + (1.03 − 0.869i)7-s + (−0.176 + 0.306i)8-s + (−0.932 + 0.339i)9-s + (0.397 + 0.689i)10-s + (−0.601 + 1.04i)11-s + (0.122 − 0.694i)12-s + (−1.02 − 0.373i)13-s + (−0.478 − 0.827i)14-s + (−1.21 − 1.02i)15-s + (0.191 + 0.160i)16-s + (−1.81 + 0.662i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.413i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.146575 + 0.677684i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.146575 + 0.677684i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.694 + 3.93i)T \) |
| 37 | \( 1 + (-5.08e3 - 6.59e3i)T \) |
good | 3 | \( 1 + (-3.82 - 21.6i)T + (-228. + 83.1i)T^{2} \) |
| 5 | \( 1 + (48.1 - 40.4i)T + (542. - 3.07e3i)T^{2} \) |
| 7 | \( 1 + (-134. + 112. i)T + (2.91e3 - 1.65e4i)T^{2} \) |
| 11 | \( 1 + (241. - 418. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (625. + 227. i)T + (2.84e5 + 2.38e5i)T^{2} \) |
| 17 | \( 1 + (2.16e3 - 789. i)T + (1.08e6 - 9.12e5i)T^{2} \) |
| 19 | \( 1 + (145. + 826. i)T + (-2.32e6 + 8.46e5i)T^{2} \) |
| 23 | \( 1 + (2.15e3 + 3.73e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (2.18e3 - 3.79e3i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 - 5.34e3T + 2.86e7T^{2} \) |
| 41 | \( 1 + (-1.96e3 - 716. i)T + (8.87e7 + 7.44e7i)T^{2} \) |
| 43 | \( 1 - 8.76e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-3.51e3 - 6.08e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (4.70e3 + 3.94e3i)T + (7.26e7 + 4.11e8i)T^{2} \) |
| 59 | \( 1 + (4.49e3 + 3.76e3i)T + (1.24e8 + 7.04e8i)T^{2} \) |
| 61 | \( 1 + (-3.26e4 - 1.18e4i)T + (6.46e8 + 5.42e8i)T^{2} \) |
| 67 | \( 1 + (5.21e4 - 4.37e4i)T + (2.34e8 - 1.32e9i)T^{2} \) |
| 71 | \( 1 + (-1.15e4 - 6.55e4i)T + (-1.69e9 + 6.17e8i)T^{2} \) |
| 73 | \( 1 + 7.14e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (2.03e4 - 1.70e4i)T + (5.34e8 - 3.03e9i)T^{2} \) |
| 83 | \( 1 + (-1.44e4 + 5.24e3i)T + (3.01e9 - 2.53e9i)T^{2} \) |
| 89 | \( 1 + (1.35e4 + 1.14e4i)T + (9.69e8 + 5.49e9i)T^{2} \) |
| 97 | \( 1 + (-1.32e4 - 2.29e4i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−14.45034626054345818166215008534, −12.87911848254529117808657755019, −11.42391558431021712726198094684, −10.64553765156254568437337130503, −10.06107042097731649767678230932, −8.517620013160368305731053864787, −7.24447298113649502266708982014, −4.58060316642954519037659125051, −4.31478097158776875003061764432, −2.56818449480577933393242694371,
0.27033145027933941987130918008, 2.19211344111157872249899527138, 4.58724213784751775439622351686, 5.94392723842167412179892773350, 7.52156927793336295413554135502, 8.092977101514814403004971715811, 9.013431781311544684570665036146, 11.50377973216242104282661312968, 12.09826563948127769367815213622, 13.25103069980080601262527796587