| L(s) = 1 | + (−0.417 − 2.04i)2-s + (−0.987 − 0.160i)3-s + (−2.17 + 0.926i)4-s + (0.956 − 0.235i)5-s + (0.0841 + 2.08i)6-s + (−1.03 + 2.17i)7-s + (0.432 + 0.626i)8-s + (0.948 + 0.316i)9-s + (−0.882 − 1.85i)10-s + (−4.33 + 1.44i)11-s + (2.29 − 0.565i)12-s + (0.533 + 3.56i)13-s + (4.88 + 1.20i)14-s + (−0.981 + 0.0792i)15-s + (−2.17 + 2.26i)16-s + (0.00896 − 0.0188i)17-s + ⋯ |
| L(s) = 1 | + (−0.295 − 1.44i)2-s + (−0.569 − 0.0926i)3-s + (−1.08 + 0.463i)4-s + (0.427 − 0.105i)5-s + (0.0343 + 0.852i)6-s + (−0.390 + 0.822i)7-s + (0.152 + 0.221i)8-s + (0.316 + 0.105i)9-s + (−0.278 − 0.587i)10-s + (−1.30 + 0.436i)11-s + (0.662 − 0.163i)12-s + (0.147 + 0.989i)13-s + (1.30 + 0.321i)14-s + (−0.253 + 0.0204i)15-s + (−0.543 + 0.565i)16-s + (0.00217 − 0.00458i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.496932 + 0.106096i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.496932 + 0.106096i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.987 + 0.160i)T \) |
| 13 | \( 1 + (-0.533 - 3.56i)T \) |
| good | 2 | \( 1 + (0.417 + 2.04i)T + (-1.83 + 0.783i)T^{2} \) |
| 5 | \( 1 + (-0.956 + 0.235i)T + (4.42 - 2.32i)T^{2} \) |
| 7 | \( 1 + (1.03 - 2.17i)T + (-4.42 - 5.42i)T^{2} \) |
| 11 | \( 1 + (4.33 - 1.44i)T + (8.79 - 6.60i)T^{2} \) |
| 17 | \( 1 + (-0.00896 + 0.0188i)T + (-10.7 - 13.1i)T^{2} \) |
| 19 | \( 1 + (-2.15 - 3.72i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.907 - 1.57i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.0433 + 0.212i)T + (-26.6 + 11.3i)T^{2} \) |
| 31 | \( 1 + (1.33 + 0.701i)T + (17.6 + 25.5i)T^{2} \) |
| 37 | \( 1 + (0.141 - 0.0897i)T + (15.8 - 33.4i)T^{2} \) |
| 41 | \( 1 + (-7.17 - 1.16i)T + (38.8 + 12.9i)T^{2} \) |
| 43 | \( 1 + (3.82 + 2.42i)T + (18.4 + 38.8i)T^{2} \) |
| 47 | \( 1 + (1.12 + 9.25i)T + (-45.6 + 11.2i)T^{2} \) |
| 53 | \( 1 + (-3.05 - 4.43i)T + (-18.7 + 49.5i)T^{2} \) |
| 59 | \( 1 + (2.07 + 2.16i)T + (-2.37 + 58.9i)T^{2} \) |
| 61 | \( 1 + (-3.11 - 0.251i)T + (60.2 + 9.78i)T^{2} \) |
| 67 | \( 1 + (9.48 + 4.03i)T + (46.4 + 48.3i)T^{2} \) |
| 71 | \( 1 + (7.37 - 9.03i)T + (-14.2 - 69.5i)T^{2} \) |
| 73 | \( 1 + (8.57 + 7.59i)T + (8.79 + 72.4i)T^{2} \) |
| 79 | \( 1 + (-1.49 - 12.2i)T + (-76.7 + 18.9i)T^{2} \) |
| 83 | \( 1 + (4.60 - 12.1i)T + (-62.1 - 55.0i)T^{2} \) |
| 89 | \( 1 + (4.14 - 7.18i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.97 - 6.82i)T + (-81.9 + 51.8i)T^{2} \) |
| 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
−11.01996192276271910833547105255, −10.08457142713282522424074787174, −9.624196051123248390413975609809, −8.701023138665579017917055652743, −7.43411091821781922390025001200, −6.13239955423773917461106200259, −5.28755158299973513924922464750, −3.95845442434610236063504889982, −2.61646955674833002505440232095, −1.68478108461108451626859201827,
0.34581064366021249484322812560, 2.92419165859191703835364999405, 4.62123686318036024810647768059, 5.60713175308774628665518272540, 6.16652127209988072963588286933, 7.27841808383915734633783547511, 7.80554444675773927191715923440, 8.871691391691171492300118597205, 9.994262434623441206596345690445, 10.54984897593754125752312477485
