| 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
−9.878933569567313047065321806314, −8.796132192006138565110092786536, −8.130926251051962482470248571822, −7.22516220964506387592500434133, −6.56591570286419007311676365105, −5.85854406306536610664757253765, −3.69434279238540776140497591675, −2.49539288041946600363551031463, −1.94200504242354233445781397338, −0.60119408425375226931567602551,
1.09199605732273445531136331248, 3.15540193641549849145753605778, 4.48057445853009175104188653272, 5.44989553898161389924253649361, 5.96838743899572245835384381132, 7.12039262068529232133606506271, 8.214307723317992595647911386097, 8.939264561832392150041284242532, 9.500756254456311840847181209553, 9.873687770512056243489434114077
