| 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 | \( 1 + 16.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.84299976679400258399318329704, −9.876867290179559204347731758891, −8.950697319250212982660946454613, −8.170232300767644860180326551693, −7.65466808206903844437966707866, −6.93360836343027251037155197415, −5.91029220523432984723106346086, −4.62649339267791949943410183176, −3.28155090946202191152525299262, −1.20068376594871024432401199450,
0.20401189596569970080406549522, 2.07029712820439294510129018655, 3.21894945255541851280507584358, 3.99652291482181082566855992379, 5.83502286717727399347200565361, 7.08045203057375520525901135785, 7.890343257052747398868905365795, 8.659196251382040363779478798460, 9.542131151995627361472630208962, 9.917980057044568162185210396640
