| L(s) = 1 | − 1.41i·2-s − 3.36i·3-s − 2.00·4-s − 4.76·6-s − 1.16·7-s + 2.82i·8-s − 2.32·9-s − 10.6i·11-s + 6.73i·12-s − 15.0i·13-s + 1.65i·14-s + 4.00·16-s − 20.0·17-s + 3.29i·18-s + 22.5i·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 1.12i·3-s − 0.500·4-s − 0.793·6-s − 0.167·7-s + 0.353i·8-s − 0.258·9-s − 0.964i·11-s + 0.560i·12-s − 1.16i·13-s + 0.118i·14-s + 0.250·16-s − 1.18·17-s + 0.183i·18-s + 1.18i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0401 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0401 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6716732514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6716732514\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + (-9.45 + 20.9i)T \) |
| good | 3 | \( 1 + 3.36iT - 9T^{2} \) |
| 7 | \( 1 + 1.16T + 49T^{2} \) |
| 11 | \( 1 + 10.6iT - 121T^{2} \) |
| 13 | \( 1 + 15.0iT - 169T^{2} \) |
| 17 | \( 1 + 20.0T + 289T^{2} \) |
| 19 | \( 1 - 22.5iT - 361T^{2} \) |
| 29 | \( 1 + 32.5T + 841T^{2} \) |
| 31 | \( 1 + 27.0T + 961T^{2} \) |
| 37 | \( 1 - 53.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 9.43T + 1.68e3T^{2} \) |
| 43 | \( 1 - 36.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 49.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 104.T + 2.80e3T^{2} \) |
| 59 | \( 1 + 53.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 23.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 59.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 55.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + 8.77iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 57.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 55.1T + 6.88e3T^{2} \) |
| 89 | \( 1 - 139. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 19.8T + 9.40e3T^{2} \) |
| 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
−8.988251713162961521539706485662, −8.075236347381641169176423603109, −7.57467066043145416123628766861, −6.34186607450892418223388490693, −5.80646553379773606701630957552, −4.49786998430120360215474574038, −3.36359758452580474138761853827, −2.41097452329276771652381193668, −1.28740608098842923319927173525, −0.20708443248906434984384753882,
1.93767939809945620008515926569, 3.47181971207101048304269453652, 4.53566269582004566559031747225, 4.75078196944605712030736068567, 6.02676668143973260120834484696, 6.98470079553490779512842607744, 7.52948723492092003020118624048, 9.003708973329775820482640520302, 9.257076227402148793389237568445, 9.865191506329610399032827913449
