| L(s) = 1 | + (−0.998 − 0.0523i)2-s + (1.80 + 2.22i)3-s + (0.994 + 0.104i)4-s + (0.724 − 2.11i)5-s + (−1.68 − 2.32i)6-s + (2.64 − 0.0482i)7-s + (−0.987 − 0.156i)8-s + (−1.08 + 5.11i)9-s + (−0.834 + 2.07i)10-s + (4.46 − 0.949i)11-s + (1.56 + 2.40i)12-s + (−3.02 − 1.54i)13-s + (−2.64 − 0.0902i)14-s + (6.02 − 2.20i)15-s + (0.978 + 0.207i)16-s + (−2.61 − 6.80i)17-s + ⋯ |
| L(s) = 1 | + (−0.706 − 0.0370i)2-s + (1.04 + 1.28i)3-s + (0.497 + 0.0522i)4-s + (0.323 − 0.946i)5-s + (−0.688 − 0.947i)6-s + (0.999 − 0.0182i)7-s + (−0.349 − 0.0553i)8-s + (−0.362 + 1.70i)9-s + (−0.263 + 0.656i)10-s + (1.34 − 0.286i)11-s + (0.450 + 0.694i)12-s + (−0.839 − 0.427i)13-s + (−0.706 − 0.0241i)14-s + (1.55 − 0.568i)15-s + (0.244 + 0.0519i)16-s + (−0.633 − 1.64i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.49058 + 0.446863i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.49058 + 0.446863i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.998 + 0.0523i)T \) |
| 5 | \( 1 + (-0.724 + 2.11i)T \) |
| 7 | \( 1 + (-2.64 + 0.0482i)T \) |
| good | 3 | \( 1 + (-1.80 - 2.22i)T + (-0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (-4.46 + 0.949i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (3.02 + 1.54i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (2.61 + 6.80i)T + (-12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (-0.199 - 1.90i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (0.292 - 5.58i)T + (-22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (0.582 - 0.801i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (3.63 - 8.16i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (8.71 - 5.66i)T + (15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (-0.915 + 0.297i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-1.12 - 1.12i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.83 + 1.08i)T + (34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (2.02 - 1.63i)T + (11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (-7.12 + 7.91i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (2.45 - 2.21i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (9.43 - 3.62i)T + (49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (8.04 + 5.84i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.79 + 7.38i)T + (-29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (1.05 + 2.37i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (1.33 - 8.45i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (0.674 + 0.748i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (0.741 + 4.68i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−11.41876794189214417185383995263, −10.30124830942715594438784036194, −9.439065539568923616341818636695, −8.999126037797064157632379672796, −8.278643155069259374894065060126, −7.19815231601374129880081850138, −5.31855253140451390934483551187, −4.56709321680805958480838575526, −3.26472831510753440783104717022, −1.69983086488623658773176688762,
1.73265869238621453181025525095, 2.32912845900100912300511585387, 3.98894665677356725831723750465, 6.15474912026924843909793945143, 6.94007843958881858767041193686, 7.57690291486580197201668546026, 8.576291128818313741394188286520, 9.215578930586327234222043869042, 10.43765262582722713552873601346, 11.41943704911704369265004521780
