L(s) = 1 | + (1.47 − 1.75i)2-s + (−1.13 − 3.10i)3-s + (−0.563 − 3.19i)4-s + (−7.11 − 2.59i)6-s + (2.54 − 1.46i)7-s + (−2.46 − 1.42i)8-s + (−6.08 + 5.10i)9-s + (0.288 − 0.500i)11-s + (−9.29 + 5.36i)12-s + (−0.229 + 0.629i)13-s + (1.16 − 6.62i)14-s + (−0.0331 + 0.0120i)16-s + (−0.226 + 0.269i)17-s + 18.1i·18-s + (3.86 + 2.02i)19-s + ⋯ |
L(s) = 1 | + (1.04 − 1.24i)2-s + (−0.653 − 1.79i)3-s + (−0.281 − 1.59i)4-s + (−2.90 − 1.05i)6-s + (0.962 − 0.555i)7-s + (−0.872 − 0.503i)8-s + (−2.02 + 1.70i)9-s + (0.0870 − 0.150i)11-s + (−2.68 + 1.54i)12-s + (−0.0635 + 0.174i)13-s + (0.312 − 1.77i)14-s + (−0.00829 + 0.00302i)16-s + (−0.0548 + 0.0653i)17-s + 4.28i·18-s + (0.886 + 0.463i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.864 - 0.502i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.864 - 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.534382 + 1.98167i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.534382 + 1.98167i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (-3.86 - 2.02i)T \) |
good | 2 | \( 1 + (-1.47 + 1.75i)T + (-0.347 - 1.96i)T^{2} \) |
| 3 | \( 1 + (1.13 + 3.10i)T + (-2.29 + 1.92i)T^{2} \) |
| 7 | \( 1 + (-2.54 + 1.46i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.288 + 0.500i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.229 - 0.629i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (0.226 - 0.269i)T + (-2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-4.05 + 0.715i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (2.30 - 1.93i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.148 + 0.257i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 8.30iT - 37T^{2} \) |
| 41 | \( 1 + (2.51 - 0.913i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (6.49 + 1.14i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (7.09 + 8.45i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-4.04 + 0.713i)T + (49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (0.467 + 0.392i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.178 - 1.01i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.14 - 2.55i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (2.29 - 13.0i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (2.44 + 6.70i)T + (-55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-1.44 + 0.527i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-11.5 + 6.65i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (6.17 + 2.24i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-7.84 + 9.34i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−11.08690720919750229030519370955, −10.16649797237016762665529051660, −8.453769983621140029059999716035, −7.56585180268747031095136830715, −6.66456024851667539504552320066, −5.49343016102556546619657847454, −4.81308371019711898865487194967, −3.22717167102056137944122054151, −1.89485075003069929844158544947, −1.11394025714478182116142967312,
3.22488241648323348905169468121, 4.27400864640059721160071378942, 5.11780462010757851093243895584, 5.42893526706020090448360616776, 6.52131753594858189772404885136, 7.78516526325792010616676774713, 8.836713002307208224661449860963, 9.645464858792168870686476526942, 10.80313249293440634208268296993, 11.49475288020500050101463708217