| L(s) = 1 | + (−1.01 − 1.68i)2-s + (2.99 − 0.225i)3-s + (0.0652 − 0.123i)4-s + (0.831 + 7.64i)5-s + (−3.41 − 4.80i)6-s + (5.33 − 1.79i)7-s + (−8.12 + 0.440i)8-s + (8.89 − 1.35i)9-s + (12.0 − 9.14i)10-s + (8.24 + 5.58i)11-s + (0.167 − 0.382i)12-s + (5.02 + 4.76i)13-s + (−8.43 − 7.16i)14-s + (4.21 + 22.6i)15-s + (8.65 + 12.7i)16-s + (1.16 − 3.44i)17-s + ⋯ |
| L(s) = 1 | + (−0.506 − 0.841i)2-s + (0.997 − 0.0753i)3-s + (0.0163 − 0.0307i)4-s + (0.166 + 1.52i)5-s + (−0.568 − 0.801i)6-s + (0.762 − 0.256i)7-s + (−1.01 + 0.0550i)8-s + (0.988 − 0.150i)9-s + (1.20 − 0.914i)10-s + (0.749 + 0.508i)11-s + (0.0139 − 0.0318i)12-s + (0.386 + 0.366i)13-s + (−0.602 − 0.511i)14-s + (0.281 + 1.51i)15-s + (0.540 + 0.797i)16-s + (0.0683 − 0.202i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.71675 - 0.598720i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.71675 - 0.598720i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.99 + 0.225i)T \) |
| 59 | \( 1 + (24.0 - 53.8i)T \) |
| good | 2 | \( 1 + (1.01 + 1.68i)T + (-1.87 + 3.53i)T^{2} \) |
| 5 | \( 1 + (-0.831 - 7.64i)T + (-24.4 + 5.37i)T^{2} \) |
| 7 | \( 1 + (-5.33 + 1.79i)T + (39.0 - 29.6i)T^{2} \) |
| 11 | \( 1 + (-8.24 - 5.58i)T + (44.7 + 112. i)T^{2} \) |
| 13 | \( 1 + (-5.02 - 4.76i)T + (9.14 + 168. i)T^{2} \) |
| 17 | \( 1 + (-1.16 + 3.44i)T + (-230. - 174. i)T^{2} \) |
| 19 | \( 1 + (2.99 + 18.2i)T + (-342. + 115. i)T^{2} \) |
| 23 | \( 1 + (28.6 - 7.96i)T + (453. - 272. i)T^{2} \) |
| 29 | \( 1 + (-21.2 + 35.3i)T + (-393. - 743. i)T^{2} \) |
| 31 | \( 1 + (-2.82 + 17.2i)T + (-910. - 306. i)T^{2} \) |
| 37 | \( 1 + (-0.226 + 4.17i)T + (-1.36e3 - 148. i)T^{2} \) |
| 41 | \( 1 + (64.7 + 17.9i)T + (1.44e3 + 866. i)T^{2} \) |
| 43 | \( 1 + (13.1 + 19.4i)T + (-684. + 1.71e3i)T^{2} \) |
| 47 | \( 1 + (8.49 - 78.0i)T + (-2.15e3 - 474. i)T^{2} \) |
| 53 | \( 1 + (-53.1 + 69.8i)T + (-751. - 2.70e3i)T^{2} \) |
| 61 | \( 1 + (38.7 - 23.2i)T + (1.74e3 - 3.28e3i)T^{2} \) |
| 67 | \( 1 + (-5.18 - 95.6i)T + (-4.46e3 + 485. i)T^{2} \) |
| 71 | \( 1 + (-4.52 + 41.6i)T + (-4.92e3 - 1.08e3i)T^{2} \) |
| 73 | \( 1 + (-0.960 + 1.13i)T + (-862. - 5.25e3i)T^{2} \) |
| 79 | \( 1 + (49.9 - 125. i)T + (-4.53e3 - 4.29e3i)T^{2} \) |
| 83 | \( 1 + (33.5 + 72.6i)T + (-4.45e3 + 5.25e3i)T^{2} \) |
| 89 | \( 1 + (-26.7 + 44.4i)T + (-3.71e3 - 6.99e3i)T^{2} \) |
| 97 | \( 1 + (8.41 + 9.90i)T + (-1.52e3 + 9.28e3i)T^{2} \) |
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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
−11.93477910417104005887746684572, −11.25929706691837052429896627950, −10.24455444020129233154284325037, −9.646597577532643137097081070330, −8.473900306592266696522036198746, −7.23953168403913951781796891170, −6.32718758825706841131002652265, −4.07544471361855079661285972756, −2.75915702691383791028333295030, −1.75322527449336209559544394574,
1.53435260134190726608545884077, 3.61903328088569385952017257171, 5.05280892306073478671519133875, 6.38700456621616829184750115876, 7.928591123434107758882210738066, 8.485382975863374962758956406566, 8.919571482007027509947869688287, 10.13273497963109950802065738932, 11.92527837743266119992185835322, 12.53923127614131479912280138152
