| L(s) = 1 | + (0.841 + 0.540i)3-s + (−1.21 + 2.65i)5-s + (0.960 + 0.281i)7-s + (0.415 + 0.909i)9-s + (−1.65 + 1.90i)11-s + (0.463 − 0.136i)13-s + (−2.45 + 1.58i)15-s + (0.446 + 3.10i)17-s + (−0.0786 + 0.547i)19-s + (0.655 + 0.756i)21-s + (3.87 − 2.82i)23-s + (−2.32 − 2.68i)25-s + (−0.142 + 0.989i)27-s + (−0.0173 − 0.120i)29-s + (6.92 − 4.45i)31-s + ⋯ |
| L(s) = 1 | + (0.485 + 0.312i)3-s + (−0.543 + 1.18i)5-s + (0.362 + 0.106i)7-s + (0.138 + 0.303i)9-s + (−0.497 + 0.574i)11-s + (0.128 − 0.0377i)13-s + (−0.635 + 0.408i)15-s + (0.108 + 0.753i)17-s + (−0.0180 + 0.125i)19-s + (0.143 + 0.165i)21-s + (0.807 − 0.589i)23-s + (−0.464 − 0.536i)25-s + (−0.0273 + 0.190i)27-s + (−0.00321 − 0.0223i)29-s + (1.24 − 0.799i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.212 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.212 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05039 + 0.846561i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05039 + 0.846561i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.841 - 0.540i)T \) |
| 23 | \( 1 + (-3.87 + 2.82i)T \) |
| good | 5 | \( 1 + (1.21 - 2.65i)T + (-3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (-0.960 - 0.281i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (1.65 - 1.90i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.463 + 0.136i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.446 - 3.10i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (0.0786 - 0.547i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (0.0173 + 0.120i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-6.92 + 4.45i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (3.95 + 8.67i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (0.578 - 1.26i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-5.21 - 3.35i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 8.69T + 47T^{2} \) |
| 53 | \( 1 + (7.41 + 2.17i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-0.227 + 0.0667i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (1.54 - 0.990i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-1.83 - 2.11i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (7.51 + 8.67i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.94 + 13.5i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (-16.3 + 4.81i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (3.34 + 7.33i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-4.15 - 2.66i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (6.49 - 14.2i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−12.00334538338554596287169212800, −10.86973851234115383893783080051, −10.45688067458579015687279066985, −9.255099541355735584793192432860, −8.080809833283115736828225752230, −7.38486310671801465991535205767, −6.24029929792996702918031552398, −4.72488487360837827852179911173, −3.54008676878843903813799259454, −2.37170007075588795597940093793,
1.08535583999476888414264669653, 3.01397133272435845321043242825, 4.46485359189169689307394961238, 5.39590740320414883658609866741, 6.95299942747799512780997656236, 8.045234435690038913346570160146, 8.604924929254661536840797739996, 9.545613671186158526310792324663, 10.86461547032228727004105129365, 11.85343305955270272944719095346
