L(s) = 1 | + (12.2 − 12.2i)7-s + 34.1i·11-s + (−25.8 − 25.8i)13-s + (15.5 + 15.5i)17-s − 32.1i·19-s + (72.9 − 72.9i)23-s + 106.·29-s + 10.3·31-s + (2.04 − 2.04i)37-s − 365. i·41-s + (10.2 + 10.2i)43-s + (260. + 260. i)47-s + 40.4i·49-s + (−42.4 + 42.4i)53-s − 375.·59-s + ⋯ |
L(s) = 1 | + (0.664 − 0.664i)7-s + 0.935i·11-s + (−0.550 − 0.550i)13-s + (0.222 + 0.222i)17-s − 0.387i·19-s + (0.661 − 0.661i)23-s + 0.681·29-s + 0.0601·31-s + (0.00907 − 0.00907i)37-s − 1.39i·41-s + (0.0363 + 0.0363i)43-s + (0.806 + 0.806i)47-s + 0.118i·49-s + (−0.109 + 0.109i)53-s − 0.828·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.994223533\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.994223533\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-12.2 + 12.2i)T - 343iT^{2} \) |
| 11 | \( 1 - 34.1iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (25.8 + 25.8i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 + (-15.5 - 15.5i)T + 4.91e3iT^{2} \) |
| 19 | \( 1 + 32.1iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-72.9 + 72.9i)T - 1.21e4iT^{2} \) |
| 29 | \( 1 - 106.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 10.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-2.04 + 2.04i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + 365. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-10.2 - 10.2i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (-260. - 260. i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 + (42.4 - 42.4i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + 375.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 770.T + 2.26e5T^{2} \) |
| 67 | \( 1 + (-603. + 603. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 + 695. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (262. + 262. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 - 32.2iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-519. + 519. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 - 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-416. + 416. i)T - 9.12e5iT^{2} \) |
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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
−9.635990473737545466858591289460, −8.745764295709011175312748140150, −7.69927229999394775277560920745, −7.25182153933796430908476687004, −6.16005870357614831394697395434, −4.92824418264949571346951997613, −4.42127643843080743251558333428, −3.07543396685561441667611246435, −1.87317392995394512805811443867, −0.58794860492295579863199310932,
1.08433006102005756221149008189, 2.34166308868588483596746630501, 3.41005299515633862952529850873, 4.67202795326786077446007929681, 5.45688613005144696667163787468, 6.35424818916889631653312092755, 7.40855211183068370164733491427, 8.268264506485625391508872967597, 8.958044244309945328061391869333, 9.790087951178280736358888246353