| L(s) = 1 | + (0.761 + 0.439i)2-s + (2.59 − 1.5i)3-s + (−3.61 − 6.25i)4-s + (−9.95 + 5.08i)5-s + 2.63·6-s + (−18.5 − 0.260i)7-s − 13.3i·8-s + (4.5 − 7.79i)9-s + (−9.81 − 0.502i)10-s + (−32.2 − 55.8i)11-s + (−18.7 − 10.8i)12-s + 73.6i·13-s + (−13.9 − 8.33i)14-s + (−18.2 + 28.1i)15-s + (−23.0 + 39.8i)16-s + (−13.4 + 7.79i)17-s + ⋯ |
| L(s) = 1 | + (0.269 + 0.155i)2-s + (0.499 − 0.288i)3-s + (−0.451 − 0.782i)4-s + (−0.890 + 0.455i)5-s + 0.179·6-s + (−0.999 − 0.0140i)7-s − 0.591i·8-s + (0.166 − 0.288i)9-s + (−0.310 − 0.0158i)10-s + (−0.883 − 1.53i)11-s + (−0.451 − 0.260i)12-s + 1.57i·13-s + (−0.266 − 0.159i)14-s + (−0.313 + 0.484i)15-s + (−0.359 + 0.623i)16-s + (−0.192 + 0.111i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.498i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.176222 - 0.660131i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.176222 - 0.660131i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.59 + 1.5i)T \) |
| 5 | \( 1 + (9.95 - 5.08i)T \) |
| 7 | \( 1 + (18.5 + 0.260i)T \) |
| good | 2 | \( 1 + (-0.761 - 0.439i)T + (4 + 6.92i)T^{2} \) |
| 11 | \( 1 + (32.2 + 55.8i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 73.6iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (13.4 - 7.79i)T + (2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-68.6 + 118. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (90.0 + 52.0i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 76.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + (48.0 + 83.2i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-211. - 122. i)T + (2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 28.9T + 6.89e4T^{2} \) |
| 43 | \( 1 + 47.5iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (100. + 58.1i)T + (5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-7.99 + 4.61i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (52.5 + 90.9i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (95.6 - 165. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (436. - 252. i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 1.08e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-398. + 229. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-8.94 + 15.4i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 691. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (-476. + 825. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 600. iT - 9.12e5T^{2} \) |
| 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
−13.30955223613784430808044998022, −11.79262345875106688402874029837, −10.74463457649831805501608035600, −9.488440726761485780874757188295, −8.519413989842635747457518421737, −7.04261123805343060564046945133, −6.09295621237789812624422173547, −4.33303126655032959013700495455, −2.97785019917386779422099456485, −0.32258332553792331994908319843,
2.94978497522349881309610208781, 3.98735124042121794617582223068, 5.26512445501543956866522602366, 7.55710087626819957110745613094, 8.021387570021474869644447906709, 9.434180807523485592177690842783, 10.35635050342149155786990036848, 12.12991822049490579187857367185, 12.63119130097775941728059560461, 13.39689594305191727621175906141
