| L(s) = 1 | + (−2.50 − 1.31i)2-s + (3.59 + 1.09i)3-s + (4.53 + 6.58i)4-s + (17.1 + 1.68i)5-s + (−7.57 − 7.46i)6-s + (15.6 + 23.4i)7-s + (−2.69 − 22.4i)8-s + (−10.6 − 7.14i)9-s + (−40.7 − 26.7i)10-s + (−28.5 + 53.4i)11-s + (9.14 + 28.6i)12-s + (−0.511 − 5.19i)13-s + (−8.36 − 79.1i)14-s + (59.8 + 24.7i)15-s + (−22.7 + 59.8i)16-s + (−80.8 + 33.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.885 − 0.465i)2-s + (0.692 + 0.210i)3-s + (0.567 + 0.823i)4-s + (1.53 + 0.151i)5-s + (−0.515 − 0.508i)6-s + (0.844 + 1.26i)7-s + (−0.119 − 0.992i)8-s + (−0.395 − 0.264i)9-s + (−1.28 − 0.847i)10-s + (−0.782 + 1.46i)11-s + (0.219 + 0.689i)12-s + (−0.0109 − 0.110i)13-s + (−0.159 − 1.51i)14-s + (1.03 + 0.426i)15-s + (−0.356 + 0.934i)16-s + (−1.15 + 0.477i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.556i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.831 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.63380 + 0.496295i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63380 + 0.496295i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.50 + 1.31i)T \) |
| good | 3 | \( 1 + (-3.59 - 1.09i)T + (22.4 + 15.0i)T^{2} \) |
| 5 | \( 1 + (-17.1 - 1.68i)T + (122. + 24.3i)T^{2} \) |
| 7 | \( 1 + (-15.6 - 23.4i)T + (-131. + 316. i)T^{2} \) |
| 11 | \( 1 + (28.5 - 53.4i)T + (-739. - 1.10e3i)T^{2} \) |
| 13 | \( 1 + (0.511 + 5.19i)T + (-2.15e3 + 428. i)T^{2} \) |
| 17 | \( 1 + (80.8 - 33.5i)T + (3.47e3 - 3.47e3i)T^{2} \) |
| 19 | \( 1 + (-73.5 + 89.5i)T + (-1.33e3 - 6.72e3i)T^{2} \) |
| 23 | \( 1 + (-2.49 - 12.5i)T + (-1.12e4 + 4.65e3i)T^{2} \) |
| 29 | \( 1 + (-137. + 73.4i)T + (1.35e4 - 2.02e4i)T^{2} \) |
| 31 | \( 1 + (-77.9 - 77.9i)T + 2.97e4iT^{2} \) |
| 37 | \( 1 + (-63.2 + 51.9i)T + (9.88e3 - 4.96e4i)T^{2} \) |
| 41 | \( 1 + (43.1 - 8.57i)T + (6.36e4 - 2.63e4i)T^{2} \) |
| 43 | \( 1 + (-349. + 106. i)T + (6.61e4 - 4.41e4i)T^{2} \) |
| 47 | \( 1 + (91.6 + 221. i)T + (-7.34e4 + 7.34e4i)T^{2} \) |
| 53 | \( 1 + (393. + 210. i)T + (8.27e4 + 1.23e5i)T^{2} \) |
| 59 | \( 1 + (-32.0 + 325. i)T + (-2.01e5 - 4.00e4i)T^{2} \) |
| 61 | \( 1 + (97.9 - 322. i)T + (-1.88e5 - 1.26e5i)T^{2} \) |
| 67 | \( 1 + (-86.9 + 286. i)T + (-2.50e5 - 1.67e5i)T^{2} \) |
| 71 | \( 1 + (-642. + 429. i)T + (1.36e5 - 3.30e5i)T^{2} \) |
| 73 | \( 1 + (-356. + 533. i)T + (-1.48e5 - 3.59e5i)T^{2} \) |
| 79 | \( 1 + (-1.24 + 3.00i)T + (-3.48e5 - 3.48e5i)T^{2} \) |
| 83 | \( 1 + (-1.03e3 - 845. i)T + (1.11e5 + 5.60e5i)T^{2} \) |
| 89 | \( 1 + (-151. + 762. i)T + (-6.51e5 - 2.69e5i)T^{2} \) |
| 97 | \( 1 + (-835. - 835. i)T + 9.12e5iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.87786713157679174456682354379, −11.82741711311553017784483855637, −10.63377278930239524668684882142, −9.543341155678273965901715856984, −9.065280167931527083663724045329, −8.007768947812103429427412181962, −6.48371986628270504231005187985, −5.02461661697332099908996237290, −2.59390168834855528315629828578, −2.07412686167218768804114584939,
1.17006198725545686406139009394, 2.59198876960664047106929408810, 5.15936989542883548340297034054, 6.24206697317345844662237555017, 7.67335437179516143367628109475, 8.431128350737017018102564151500, 9.468844443177287168439621251287, 10.55424208612939586184438797589, 11.19413182246290905500155622046, 13.38392383665399372877161124765
