| L(s) = 1 | + 0.513i·2-s + (−2.25 + 1.97i)3-s + 3.73·4-s + (−6.08 + 3.51i)5-s + (−1.01 − 1.16i)6-s + (1.23 + 6.88i)7-s + 3.97i·8-s + (1.21 − 8.91i)9-s + (−1.80 − 3.12i)10-s + (3.15 + 1.82i)11-s + (−8.44 + 7.37i)12-s + (−3.79 + 6.56i)13-s + (−3.53 + 0.636i)14-s + (6.81 − 19.9i)15-s + 12.9·16-s + (17.5 − 10.1i)17-s + ⋯ |
| L(s) = 1 | + 0.256i·2-s + (−0.753 + 0.657i)3-s + 0.934·4-s + (−1.21 + 0.702i)5-s + (−0.168 − 0.193i)6-s + (0.177 + 0.984i)7-s + 0.496i·8-s + (0.134 − 0.990i)9-s + (−0.180 − 0.312i)10-s + (0.287 + 0.165i)11-s + (−0.703 + 0.614i)12-s + (−0.291 + 0.505i)13-s + (−0.252 + 0.0454i)14-s + (0.454 − 1.32i)15-s + 0.806·16-s + (1.03 − 0.594i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.582921 + 0.774899i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.582921 + 0.774899i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.25 - 1.97i)T \) |
| 7 | \( 1 + (-1.23 - 6.88i)T \) |
| good | 2 | \( 1 - 0.513iT - 4T^{2} \) |
| 5 | \( 1 + (6.08 - 3.51i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-3.15 - 1.82i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (3.79 - 6.56i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (-17.5 + 10.1i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-13.6 + 23.7i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-3.42 + 1.97i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (23.7 - 13.6i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 - 4.85T + 961T^{2} \) |
| 37 | \( 1 + (18.7 - 32.4i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-61.1 - 35.2i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (9.41 + 16.3i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + 23.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-23.1 + 13.3i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + 52.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 106.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 102.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 138. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (34.7 + 60.2i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 - 23.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + (25.9 - 14.9i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-135. - 78.3i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-2.93 - 5.07i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
| 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
−15.25804263263977551417283190885, −14.58969898244636939671587026426, −12.13581119498476321502143533852, −11.66631022843912218683924029320, −10.89767598628718036900631125092, −9.361709123769738505915741265613, −7.61910355865884830188917194826, −6.57614481307445689678508341184, −5.09688150166375551874556813437, −3.18118459213150800443914694071,
1.08356770092452054656093718082, 3.86025513523402039957113322984, 5.73661583747264119991727001480, 7.41274258466812957685862299229, 7.87072575415999957983883325061, 10.25174446219267997697713202033, 11.24881099276213474565613988536, 12.11776313793675459379406471915, 12.76922024180126195561830225489, 14.38325269697503625852137423852
