| L(s) = 1 | + (−0.415 − 0.909i)2-s + 2.02·3-s + (−0.654 + 0.755i)4-s + (3.72 − 1.09i)5-s + (−0.842 − 1.84i)6-s + (−0.174 + 1.21i)7-s + (0.959 + 0.281i)8-s + 1.10·9-s + (−2.54 − 2.93i)10-s + (−2.78 + 1.79i)11-s + (−1.32 + 1.53i)12-s + (−3.07 + 3.55i)13-s + (1.17 − 0.344i)14-s + (7.55 − 2.21i)15-s + (−0.142 − 0.989i)16-s + (2.43 − 1.56i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 − 0.643i)2-s + 1.17·3-s + (−0.327 + 0.377i)4-s + (1.66 − 0.489i)5-s + (−0.343 − 0.752i)6-s + (−0.0657 + 0.457i)7-s + (0.339 + 0.0996i)8-s + 0.369·9-s + (−0.804 − 0.928i)10-s + (−0.840 + 0.541i)11-s + (−0.383 + 0.442i)12-s + (−0.853 + 0.984i)13-s + (0.313 − 0.0920i)14-s + (1.95 − 0.573i)15-s + (−0.0355 − 0.247i)16-s + (0.591 − 0.380i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.765 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.765 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.59924 - 0.582329i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59924 - 0.582329i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 + 0.909i)T \) |
| 11 | \( 1 + (2.78 - 1.79i)T \) |
| good | 3 | \( 1 - 2.02T + 3T^{2} \) |
| 5 | \( 1 + (-3.72 + 1.09i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (0.174 - 1.21i)T + (-6.71 - 1.97i)T^{2} \) |
| 13 | \( 1 + (3.07 - 3.55i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.43 + 1.56i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (5.30 + 3.41i)T + (7.89 + 17.2i)T^{2} \) |
| 23 | \( 1 + (0.356 + 2.47i)T + (-22.0 + 6.47i)T^{2} \) |
| 29 | \( 1 + (-0.0294 - 0.0189i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (3.84 + 4.43i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (0.824 + 0.951i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (1.79 + 3.93i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-8.29 - 2.43i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + (-1.01 + 2.21i)T + (-30.7 - 35.5i)T^{2} \) |
| 53 | \( 1 + (1.08 - 7.55i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (4.78 - 10.4i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-1.54 + 3.38i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (3.81 + 8.34i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (-12.9 - 8.29i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.03 - 7.23i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (3.80 - 1.11i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (0.777 - 5.40i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (6.78 - 4.36i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-17.4 - 5.11i)T + (81.6 + 52.4i)T^{2} \) |
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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
−12.30711237261132303227615745225, −10.77552570802247357608436014168, −9.732911602002415426626047961838, −9.269170652115222923992450571864, −8.556545586330547044451799789392, −7.25431830034381634327977029017, −5.71982029036449411801970928873, −4.51566553632643374412495337410, −2.52037552500105933506139557957, −2.15875101975986093056318812733,
2.09560386729759348832273697565, 3.29502133121020374416536011154, 5.31682779740965523934648816942, 6.13786112945106536851462569651, 7.45078293051865304198440680707, 8.267882623021851899630469447012, 9.278497786655750396972645525823, 10.18325426371076742528987834059, 10.60178435372927918706254556728, 12.79218893669196693803589780843
