| L(s) = 1 | + (−0.933 + 0.358i)2-s + (−0.647 + 0.420i)3-s + (0.743 − 0.669i)4-s + (1.72 + 1.42i)5-s + (0.453 − 0.624i)6-s + (−2.58 − 0.552i)7-s + (−0.453 + 0.891i)8-s + (−0.977 + 2.19i)9-s + (−2.12 − 0.709i)10-s + (−3.05 − 1.28i)11-s + (−0.199 + 0.745i)12-s + (1.02 − 6.49i)13-s + (2.61 − 0.411i)14-s + (−1.71 − 0.195i)15-s + (0.104 − 0.994i)16-s + (−2.69 − 1.03i)17-s + ⋯ |
| L(s) = 1 | + (−0.660 + 0.253i)2-s + (−0.373 + 0.242i)3-s + (0.371 − 0.334i)4-s + (0.771 + 0.636i)5-s + (0.185 − 0.254i)6-s + (−0.977 − 0.208i)7-s + (−0.160 + 0.315i)8-s + (−0.325 + 0.732i)9-s + (−0.670 − 0.224i)10-s + (−0.921 − 0.387i)11-s + (−0.0576 + 0.215i)12-s + (0.285 − 1.80i)13-s + (0.698 − 0.110i)14-s + (−0.442 − 0.0504i)15-s + (0.0261 − 0.248i)16-s + (−0.654 − 0.251i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.383529 - 0.317087i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.383529 - 0.317087i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.933 - 0.358i)T \) |
| 5 | \( 1 + (-1.72 - 1.42i)T \) |
| 7 | \( 1 + (2.58 + 0.552i)T \) |
| 11 | \( 1 + (3.05 + 1.28i)T \) |
| good | 3 | \( 1 + (0.647 - 0.420i)T + (1.22 - 2.74i)T^{2} \) |
| 13 | \( 1 + (-1.02 + 6.49i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (2.69 + 1.03i)T + (12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (-2.93 + 3.25i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (-0.00532 + 0.0198i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (1.52 + 0.494i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.795 + 0.0836i)T + (30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-2.22 + 3.42i)T + (-15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (-1.90 + 0.618i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (5.26 + 5.26i)T + 43iT^{2} \) |
| 47 | \( 1 + (6.84 - 0.358i)T + (46.7 - 4.91i)T^{2} \) |
| 53 | \( 1 + (-2.61 + 3.22i)T + (-11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (1.05 + 1.16i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-4.25 - 0.447i)T + (59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (4.05 + 15.1i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-5.15 - 3.74i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-14.2 - 0.748i)T + (72.6 + 7.63i)T^{2} \) |
| 79 | \( 1 + (-0.710 + 1.59i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (8.67 - 1.37i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (3.35 + 5.81i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (16.9 + 2.67i)T + (92.2 + 29.9i)T^{2} \) |
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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
−10.15097024854473499150996275773, −9.520760275138392212866526443723, −8.386665005425717004416272650182, −7.55982253624725531559025266285, −6.63441593146073166588607790306, −5.70098461584419762683907828032, −5.19914943887937896775777365070, −3.22403112138058047673104035580, −2.46978841287083087758023757593, −0.32280257890928349957776245247,
1.42848543201294534638259686107, 2.62797530900249150740889251332, 4.00106615871912445345829815081, 5.34515925466295058057971459697, 6.35759588455267652013854557504, 6.79576983458577282635207427636, 8.147412744487970658028554481579, 9.112287554690247562207088931482, 9.518933307726922458368738806224, 10.26190320762501391955776534538
