L(s) = 1 | + (0.5 + 0.866i)2-s + (0.221 + 0.383i)3-s + (−0.499 + 0.866i)4-s + (0.445 + 0.772i)5-s + (−0.221 + 0.383i)6-s + 2.52·7-s − 0.999·8-s + (1.40 − 2.42i)9-s + (−0.445 + 0.772i)10-s + 1.95·11-s − 0.442·12-s + (3.22 − 5.59i)13-s + (1.26 + 2.18i)14-s + (−0.197 + 0.341i)15-s + (−0.5 − 0.866i)16-s + (1.71 + 2.96i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.127 + 0.221i)3-s + (−0.249 + 0.433i)4-s + (0.199 + 0.345i)5-s + (−0.0903 + 0.156i)6-s + 0.952·7-s − 0.353·8-s + (0.467 − 0.809i)9-s + (−0.140 + 0.244i)10-s + 0.588·11-s − 0.127·12-s + (0.895 − 1.55i)13-s + (0.336 + 0.583i)14-s + (−0.0509 + 0.0882i)15-s + (−0.125 − 0.216i)16-s + (0.415 + 0.718i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 - 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.519 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.97688 + 1.11180i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.97688 + 1.11180i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.221 - 0.383i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.445 - 0.772i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 - 2.52T + 7T^{2} \) |
| 11 | \( 1 - 1.95T + 11T^{2} \) |
| 13 | \( 1 + (-3.22 + 5.59i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.71 - 2.96i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (4.09 - 7.08i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.29 - 3.96i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 8.79T + 31T^{2} \) |
| 37 | \( 1 + 5.97T + 37T^{2} \) |
| 41 | \( 1 + (-1.74 - 3.02i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.12 - 5.40i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.27 + 9.13i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.88 - 3.25i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.42 + 2.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.22 + 2.12i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.33 + 2.31i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.0282 + 0.0488i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.48 - 6.03i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.56 - 7.90i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + (0.933 - 1.61i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.91 + 6.77i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.55361586617873425003509936342, −9.629107775263755609357147235888, −8.647418410542743251988503106783, −7.933032026531586279916530607560, −7.03919039252889841292563623934, −5.98038209080096317612001323226, −5.34560348001099619698908866577, −3.97895459892587577048609887508, −3.35762292552573694488109964729, −1.45770859958325146078106448008,
1.43802281331933286445902747949, 2.17090470355986532715872659854, 3.90520899979445098405627888533, 4.60894489128934444182065278815, 5.56913010685035324371332483587, 6.73838607886392925784348965324, 7.69987067531844872004608057808, 8.783090833533138817219764517988, 9.302939119682947097492505974069, 10.51693806591608014183675090467