L(s) = 1 | + (0.100 + 1.41i)2-s + (−0.429 − 0.0934i)3-s + (−1.97 + 0.284i)4-s + (−2.14 + 4.51i)5-s + (0.0884 − 0.615i)6-s + (−0.202 + 0.371i)7-s + (−0.601 − 2.76i)8-s + (−8.01 − 3.65i)9-s + (−6.58 − 2.57i)10-s + (−2.84 − 3.28i)11-s + (0.876 + 0.0627i)12-s + (−0.583 − 1.06i)13-s + (−0.544 − 0.248i)14-s + (1.34 − 1.73i)15-s + (3.83 − 1.12i)16-s + (−2.86 − 3.82i)17-s + ⋯ |
L(s) = 1 | + (0.0504 + 0.705i)2-s + (−0.143 − 0.0311i)3-s + (−0.494 + 0.0711i)4-s + (−0.429 + 0.902i)5-s + (0.0147 − 0.102i)6-s + (−0.0289 + 0.0530i)7-s + (−0.0751 − 0.345i)8-s + (−0.890 − 0.406i)9-s + (−0.658 − 0.257i)10-s + (−0.258 − 0.298i)11-s + (0.0730 + 0.00522i)12-s + (−0.0448 − 0.0821i)13-s + (−0.0388 − 0.0177i)14-s + (0.0896 − 0.115i)15-s + (0.239 − 0.0704i)16-s + (−0.168 − 0.225i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 + 0.763i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.646 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0672222 - 0.144962i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0672222 - 0.144962i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.100 - 1.41i)T \) |
| 5 | \( 1 + (2.14 - 4.51i)T \) |
| 23 | \( 1 + (21.7 + 7.56i)T \) |
good | 3 | \( 1 + (0.429 + 0.0934i)T + (8.18 + 3.73i)T^{2} \) |
| 7 | \( 1 + (0.202 - 0.371i)T + (-26.4 - 41.2i)T^{2} \) |
| 11 | \( 1 + (2.84 + 3.28i)T + (-17.2 + 119. i)T^{2} \) |
| 13 | \( 1 + (0.583 + 1.06i)T + (-91.3 + 142. i)T^{2} \) |
| 17 | \( 1 + (2.86 + 3.82i)T + (-81.4 + 277. i)T^{2} \) |
| 19 | \( 1 + (8.41 - 1.21i)T + (346. - 101. i)T^{2} \) |
| 29 | \( 1 + (45.4 + 6.54i)T + (806. + 236. i)T^{2} \) |
| 31 | \( 1 + (-17.0 - 10.9i)T + (399. + 874. i)T^{2} \) |
| 37 | \( 1 + (6.00 + 16.1i)T + (-1.03e3 + 896. i)T^{2} \) |
| 41 | \( 1 + (-24.7 - 54.1i)T + (-1.10e3 + 1.27e3i)T^{2} \) |
| 43 | \( 1 + (-2.63 - 0.572i)T + (1.68e3 + 768. i)T^{2} \) |
| 47 | \( 1 + (11.9 - 11.9i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (21.7 + 11.8i)T + (1.51e3 + 2.36e3i)T^{2} \) |
| 59 | \( 1 + (17.6 - 60.1i)T + (-2.92e3 - 1.88e3i)T^{2} \) |
| 61 | \( 1 + (30.3 + 19.5i)T + (1.54e3 + 3.38e3i)T^{2} \) |
| 67 | \( 1 + (-4.86 - 68.0i)T + (-4.44e3 + 638. i)T^{2} \) |
| 71 | \( 1 + (43.2 - 49.9i)T + (-717. - 4.98e3i)T^{2} \) |
| 73 | \( 1 + (-68.5 + 91.5i)T + (-1.50e3 - 5.11e3i)T^{2} \) |
| 79 | \( 1 + (39.7 - 135. i)T + (-5.25e3 - 3.37e3i)T^{2} \) |
| 83 | \( 1 + (41.1 - 15.3i)T + (5.20e3 - 4.51e3i)T^{2} \) |
| 89 | \( 1 + (-11.0 - 17.2i)T + (-3.29e3 + 7.20e3i)T^{2} \) |
| 97 | \( 1 + (17.6 + 6.56i)T + (7.11e3 + 6.16e3i)T^{2} \) |
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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
−12.58787117307283412149739581449, −11.56354168488957760872419687574, −10.77616000950155343398084840691, −9.591063397023615663918014669167, −8.441290292959110146887254776060, −7.58175352381378462781617286521, −6.45534348917860202046876395097, −5.69324572904019295705284179293, −4.11851066193286911254202906676, −2.82731719250544667862630815661,
0.081957889718398054077298435990, 2.03655270468918547105788207348, 3.72193650748477973738140531723, 4.86664639442678520467219317894, 5.86638289883811247734150547105, 7.63823151519376518569312693803, 8.535356301459869557831188067747, 9.420533403637225510600287702305, 10.57591101430591920075582763240, 11.46763356216093848891236072071