| L(s) = 1 | + (−1.98 − 1.35i)2-s + (0.272 + 0.252i)3-s + (1.37 + 3.49i)4-s + (−2.27 + 0.701i)5-s + (−0.198 − 0.868i)6-s + (−1.25 + 2.32i)7-s + (0.936 − 4.10i)8-s + (−0.213 − 2.85i)9-s + (5.45 + 1.68i)10-s + (−0.208 + 2.77i)11-s + (−0.509 + 1.29i)12-s + (0.900 − 0.433i)13-s + (5.63 − 2.91i)14-s + (−0.795 − 0.383i)15-s + (−1.89 + 1.75i)16-s + (0.248 − 0.0374i)17-s + ⋯ |
| L(s) = 1 | + (−1.40 − 0.955i)2-s + (0.157 + 0.145i)3-s + (0.685 + 1.74i)4-s + (−1.01 + 0.313i)5-s + (−0.0808 − 0.354i)6-s + (−0.474 + 0.880i)7-s + (0.330 − 1.45i)8-s + (−0.0712 − 0.951i)9-s + (1.72 + 0.532i)10-s + (−0.0628 + 0.838i)11-s + (−0.146 + 0.374i)12-s + (0.249 − 0.120i)13-s + (1.50 − 0.779i)14-s + (−0.205 − 0.0989i)15-s + (−0.473 + 0.439i)16-s + (0.0602 − 0.00907i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 + 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.903 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0468051 - 0.208275i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0468051 - 0.208275i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (1.25 - 2.32i)T \) |
| 13 | \( 1 + (-0.900 + 0.433i)T \) |
| good | 2 | \( 1 + (1.98 + 1.35i)T + (0.730 + 1.86i)T^{2} \) |
| 3 | \( 1 + (-0.272 - 0.252i)T + (0.224 + 2.99i)T^{2} \) |
| 5 | \( 1 + (2.27 - 0.701i)T + (4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (0.208 - 2.77i)T + (-10.8 - 1.63i)T^{2} \) |
| 17 | \( 1 + (-0.248 + 0.0374i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (-1.18 - 2.05i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.18 + 0.178i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (5.16 + 6.48i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (-1.55 + 2.69i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.822 + 2.09i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (-2.43 + 10.6i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-0.474 - 2.07i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (3.75 + 2.55i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (3.22 + 8.21i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (3.85 + 1.18i)T + (48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (-1.23 + 3.14i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (6.14 - 10.6i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.14 + 1.42i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-4.18 + 2.85i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (2.13 + 3.69i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.41 + 3.09i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (-0.367 - 4.90i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−9.917980057044568162185210396640, −9.542131151995627361472630208962, −8.659196251382040363779478798460, −7.890343257052747398868905365795, −7.08045203057375520525901135785, −5.83502286717727399347200565361, −3.99652291482181082566855992379, −3.21894945255541851280507584358, −2.07029712820439294510129018655, −0.20401189596569970080406549522,
1.20068376594871024432401199450, 3.28155090946202191152525299262, 4.62649339267791949943410183176, 5.91029220523432984723106346086, 6.93360836343027251037155197415, 7.65466808206903844437966707866, 8.170232300767644860180326551693, 8.950697319250212982660946454613, 9.876867290179559204347731758891, 10.84299976679400258399318329704
