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
−10.51693806591608014183675090467, −9.302939119682947097492505974069, −8.783090833533138817219764517988, −7.69987067531844872004608057808, −6.73838607886392925784348965324, −5.56913010685035324371332483587, −4.60894489128934444182065278815, −3.90520899979445098405627888533, −2.17090470355986532715872659854, −1.43802281331933286445902747949,
1.45770859958325146078106448008, 3.35762292552573694488109964729, 3.97895459892587577048609887508, 5.34560348001099619698908866577, 5.98038209080096317612001323226, 7.03919039252889841292563623934, 7.933032026531586279916530607560, 8.647418410542743251988503106783, 9.629107775263755609357147235888, 10.55361586617873425003509936342