| L(s) = 1 | + (2.24 + 3.35i)3-s + (−0.109 + 0.552i)5-s + (1.03 + 5.19i)7-s + (−2.79 + 6.74i)9-s + (4.88 + 3.26i)11-s + (−4.17 − 4.17i)13-s + (−2.10 + 0.871i)15-s + (−11.8 + 12.1i)17-s + (−0.379 − 0.916i)19-s + (−15.1 + 15.1i)21-s + (12.9 − 19.3i)23-s + (22.8 + 9.44i)25-s + (6.70 − 1.33i)27-s + (−13.8 − 2.75i)29-s + (39.4 − 26.3i)31-s + ⋯ |
| L(s) = 1 | + (0.747 + 1.11i)3-s + (−0.0219 + 0.110i)5-s + (0.147 + 0.742i)7-s + (−0.310 + 0.749i)9-s + (0.444 + 0.296i)11-s + (−0.320 − 0.320i)13-s + (−0.140 + 0.0580i)15-s + (−0.699 + 0.714i)17-s + (−0.0199 − 0.0482i)19-s + (−0.720 + 0.720i)21-s + (0.560 − 0.839i)23-s + (0.912 + 0.377i)25-s + (0.248 − 0.0494i)27-s + (−0.478 − 0.0951i)29-s + (1.27 − 0.851i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.33694 + 1.14574i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33694 + 1.14574i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (11.8 - 12.1i)T \) |
| good | 3 | \( 1 + (-2.24 - 3.35i)T + (-3.44 + 8.31i)T^{2} \) |
| 5 | \( 1 + (0.109 - 0.552i)T + (-23.0 - 9.56i)T^{2} \) |
| 7 | \( 1 + (-1.03 - 5.19i)T + (-45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (-4.88 - 3.26i)T + (46.3 + 111. i)T^{2} \) |
| 13 | \( 1 + (4.17 + 4.17i)T + 169iT^{2} \) |
| 19 | \( 1 + (0.379 + 0.916i)T + (-255. + 255. i)T^{2} \) |
| 23 | \( 1 + (-12.9 + 19.3i)T + (-202. - 488. i)T^{2} \) |
| 29 | \( 1 + (13.8 + 2.75i)T + (776. + 321. i)T^{2} \) |
| 31 | \( 1 + (-39.4 + 26.3i)T + (367. - 887. i)T^{2} \) |
| 37 | \( 1 + (15.6 + 23.4i)T + (-523. + 1.26e3i)T^{2} \) |
| 41 | \( 1 + (2.15 + 10.8i)T + (-1.55e3 + 643. i)T^{2} \) |
| 43 | \( 1 + (-2.23 + 5.39i)T + (-1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (32.6 + 32.6i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (37.6 + 90.9i)T + (-1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-45.2 - 18.7i)T + (2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (20.7 - 4.12i)T + (3.43e3 - 1.42e3i)T^{2} \) |
| 67 | \( 1 - 122. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (19.8 + 29.6i)T + (-1.92e3 + 4.65e3i)T^{2} \) |
| 73 | \( 1 + (-19.1 + 96.2i)T + (-4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-100. - 67.0i)T + (2.38e3 + 5.76e3i)T^{2} \) |
| 83 | \( 1 + (91.2 - 37.7i)T + (4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (50.1 - 50.1i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (-12.2 - 2.44i)T + (8.69e3 + 3.60e3i)T^{2} \) |
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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
−13.27538272340185258637543078829, −12.18405657523024803757396853913, −10.96586236606865404368041824331, −9.988506067051127062745026972166, −9.039782910123580063416575971454, −8.317054701766907955390478038523, −6.67539458566368577166208125676, −5.10435856649054630543805328226, −3.93717103824967632094587258492, −2.52885899060700950228444843906,
1.28511381143449195310747397612, 2.95594056294927074422667597061, 4.67276764283099317941509855395, 6.58709678879624519521551445185, 7.33237901479472267238105121301, 8.384420750078140278534029164313, 9.381798671697046851213443352110, 10.81254123820630682868114732905, 11.92099229997690037614448663665, 12.93597213022772066067370447157
