| L(s) = 1 | + (0.640 + 1.97i)2-s + (1.43 + 1.58i)3-s + (−1.85 + 1.34i)4-s + (−1.17 − 2.03i)5-s + (−2.21 + 3.83i)6-s + (0.384 + 3.65i)7-s + (−0.492 − 0.357i)8-s + (−0.164 + 1.56i)9-s + (3.25 − 3.61i)10-s + (3.91 + 1.74i)11-s + (−4.79 − 1.02i)12-s + (−2.04 + 0.433i)13-s + (−6.95 + 3.09i)14-s + (1.55 − 4.77i)15-s + (−1.02 + 3.16i)16-s + (−1.94 + 0.865i)17-s + ⋯ |
| L(s) = 1 | + (0.452 + 1.39i)2-s + (0.826 + 0.917i)3-s + (−0.927 + 0.674i)4-s + (−0.525 − 0.909i)5-s + (−0.904 + 1.56i)6-s + (0.145 + 1.38i)7-s + (−0.174 − 0.126i)8-s + (−0.0549 + 0.522i)9-s + (1.02 − 1.14i)10-s + (1.17 + 0.524i)11-s + (−1.38 − 0.294i)12-s + (−0.566 + 0.120i)13-s + (−1.85 + 0.827i)14-s + (0.400 − 1.23i)15-s + (−0.256 + 0.790i)16-s + (−0.471 + 0.209i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.103734 - 2.44687i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.103734 - 2.44687i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.640 - 1.97i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (-1.43 - 1.58i)T + (-0.313 + 2.98i)T^{2} \) |
| 5 | \( 1 + (1.17 + 2.03i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.384 - 3.65i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (-3.91 - 1.74i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (2.04 - 0.433i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (1.94 - 0.865i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-0.606 - 0.128i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (2.71 + 1.97i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.425 - 1.31i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-0.137 + 0.237i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.86 - 3.17i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (-0.263 - 0.0560i)T + (39.2 + 17.4i)T^{2} \) |
| 47 | \( 1 + (-1.66 + 5.11i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.993 + 9.45i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (3.89 + 4.33i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + 2.22T + 61T^{2} \) |
| 67 | \( 1 + (-6.80 - 11.7i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.139 + 1.32i)T + (-69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (-12.9 - 5.76i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (7.92 - 3.52i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-3.46 + 3.85i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (-4.05 + 2.94i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (5.43 - 3.94i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.987846511127170880922743472762, −9.207514808900393819002883436620, −8.584461295873678208256401558286, −8.246974636913438155704033008745, −7.00962300186521237044268386095, −6.14005884483465353245860895687, −5.06214633024320067812502339790, −4.52447908830167970159316261963, −3.70868943224482631000848731218, −2.14381542870162967989931803122,
0.944572577765009510414937832097, 2.04909472234466268002809210030, 3.14317343410380338739218566782, 3.72893491725405204229276278126, 4.62720038030374165432813697690, 6.44161122604247426605135411661, 7.32792846599374551306721935078, 7.62253667175466583256790434463, 8.905573938628021009733504283023, 9.855592012314927344338237249468
