| L(s) = 1 | + (−1.86 + 1.35i)2-s + (−0.265 + 2.52i)3-s + (1.02 − 3.16i)4-s + (1.24 − 2.16i)5-s + (−2.93 − 5.08i)6-s + (−1.56 + 0.333i)7-s + (0.944 + 2.90i)8-s + (−3.38 − 0.720i)9-s + (0.602 + 5.73i)10-s + (−0.490 − 0.545i)11-s + (7.72 + 3.43i)12-s + (1.73 − 0.771i)13-s + (2.47 − 2.74i)14-s + (5.13 + 3.73i)15-s + (−0.326 − 0.237i)16-s + (2.93 − 3.25i)17-s + ⋯ |
| L(s) = 1 | + (−1.32 + 0.959i)2-s + (−0.153 + 1.46i)3-s + (0.513 − 1.58i)4-s + (0.558 − 0.967i)5-s + (−1.19 − 2.07i)6-s + (−0.592 + 0.125i)7-s + (0.333 + 1.02i)8-s + (−1.12 − 0.240i)9-s + (0.190 + 1.81i)10-s + (−0.148 − 0.164i)11-s + (2.22 + 0.992i)12-s + (0.480 − 0.213i)13-s + (0.661 − 0.734i)14-s + (1.32 + 0.964i)15-s + (−0.0817 − 0.0593i)16-s + (0.710 − 0.789i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 - 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.155942 + 0.658337i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.155942 + 0.658337i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 + (1.86 - 1.35i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (0.265 - 2.52i)T + (-2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 + (-1.24 + 2.16i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (1.56 - 0.333i)T + (6.39 - 2.84i)T^{2} \) |
| 11 | \( 1 + (0.490 + 0.545i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-1.73 + 0.771i)T + (8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (-2.93 + 3.25i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (4.23 + 1.88i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (-2.19 - 6.77i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (0.104 - 0.0757i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (-4.21 - 7.30i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.770 - 7.33i)T + (-40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (0.210 + 0.0937i)T + (28.7 + 31.9i)T^{2} \) |
| 47 | \( 1 + (-6.50 - 4.72i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-5.60 - 1.19i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (0.993 - 9.45i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 - 7.84T + 61T^{2} \) |
| 67 | \( 1 + (2.41 - 4.17i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.33 - 0.707i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (1.80 + 2.00i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (3.02 - 3.36i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-0.279 - 2.65i)T + (-81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 + (-0.681 + 2.09i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-3.79 + 11.6i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−9.873687770512056243489434114077, −9.500756254456311840847181209553, −8.939264561832392150041284242532, −8.214307723317992595647911386097, −7.12039262068529232133606506271, −5.96838743899572245835384381132, −5.44989553898161389924253649361, −4.48057445853009175104188653272, −3.15540193641549849145753605778, −1.09199605732273445531136331248,
0.60119408425375226931567602551, 1.94200504242354233445781397338, 2.49539288041946600363551031463, 3.69434279238540776140497591675, 5.85854406306536610664757253765, 6.56591570286419007311676365105, 7.22516220964506387592500434133, 8.130926251051962482470248571822, 8.796132192006138565110092786536, 9.878933569567313047065321806314
