| L(s) = 1 | + (−1.10 + 0.878i)2-s + (−0.117 − 1.72i)3-s + (0.457 − 1.94i)4-s + (3.15 + 1.68i)5-s + (1.64 + 1.81i)6-s + (4.43 + 0.882i)7-s + (1.20 + 2.55i)8-s + (−2.97 + 0.404i)9-s + (−4.98 + 0.902i)10-s + (0.254 − 0.208i)11-s + (−3.41 − 0.562i)12-s + (−0.958 + 0.512i)13-s + (−5.69 + 2.91i)14-s + (2.54 − 5.65i)15-s + (−3.58 − 1.78i)16-s + (−3.78 + 1.56i)17-s + ⋯ |
| L(s) = 1 | + (−0.783 + 0.621i)2-s + (−0.0676 − 0.997i)3-s + (0.228 − 0.973i)4-s + (1.41 + 0.754i)5-s + (0.672 + 0.739i)6-s + (1.67 + 0.333i)7-s + (0.425 + 0.905i)8-s + (−0.990 + 0.134i)9-s + (−1.57 + 0.285i)10-s + (0.0766 − 0.0629i)11-s + (−0.986 − 0.162i)12-s + (−0.265 + 0.142i)13-s + (−1.52 + 0.780i)14-s + (0.657 − 1.45i)15-s + (−0.895 − 0.445i)16-s + (−0.918 + 0.380i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.286i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27541 + 0.186940i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27541 + 0.186940i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.10 - 0.878i)T \) |
| 3 | \( 1 + (0.117 + 1.72i)T \) |
| good | 5 | \( 1 + (-3.15 - 1.68i)T + (2.77 + 4.15i)T^{2} \) |
| 7 | \( 1 + (-4.43 - 0.882i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-0.254 + 0.208i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (0.958 - 0.512i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (3.78 - 1.56i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.753 - 2.48i)T + (-15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (-3.81 - 2.55i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (0.997 + 0.818i)T + (5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (-4.20 + 4.20i)T - 31iT^{2} \) |
| 37 | \( 1 + (0.965 + 3.18i)T + (-30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (7.88 + 5.26i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-1.17 + 11.9i)T + (-42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (0.396 - 0.164i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (4.10 - 3.36i)T + (10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (12.3 + 6.58i)T + (32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (-0.222 - 2.25i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (2.37 - 0.234i)T + (65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (-11.7 - 2.33i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (2.36 + 11.9i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (2.34 - 5.65i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-1.54 + 5.09i)T + (-69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (-3.82 - 5.72i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-10.5 - 10.5i)T + 97iT^{2} \) |
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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
−11.08721990406542656761763474685, −10.58820160132796929688126847747, −9.309478681201167625187138219384, −8.516948789851923360946738972374, −7.60731769474315518049169443647, −6.71934241788550177384896471281, −5.87165245882207482425039497251, −5.09188506124775770702229712140, −2.25387694992676221357971000954, −1.67047808150816220049916358366,
1.39806354031140102665530547233, 2.68532587722620071695132681042, 4.62933924059802367374567659623, 4.94368802280775279025702899023, 6.55698024512301874528407053768, 8.097272073896330759802111670630, 8.814394434529966688971898147212, 9.480202839283273745429539941734, 10.37314633963014108732876775901, 11.04268494528162943143122129433
