| L(s) = 1 | + (0.939 − 0.342i)2-s + (−0.173 + 0.984i)3-s + (0.766 − 0.642i)4-s + (0.907 + 0.761i)5-s + (0.173 + 0.984i)6-s + (−0.266 − 0.460i)7-s + (0.500 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (1.11 + 0.405i)10-s + (−0.939 + 1.62i)11-s + (0.5 + 0.866i)12-s + (−0.673 − 3.82i)13-s + (−0.407 − 0.342i)14-s + (−0.907 + 0.761i)15-s + (0.173 − 0.984i)16-s + (−1.09 + 0.397i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (−0.100 + 0.568i)3-s + (0.383 − 0.321i)4-s + (0.405 + 0.340i)5-s + (0.0708 + 0.402i)6-s + (−0.100 − 0.174i)7-s + (0.176 − 0.306i)8-s + (−0.313 − 0.114i)9-s + (0.352 + 0.128i)10-s + (−0.283 + 0.490i)11-s + (0.144 + 0.249i)12-s + (−0.186 − 1.05i)13-s + (−0.108 − 0.0914i)14-s + (−0.234 + 0.196i)15-s + (0.0434 − 0.246i)16-s + (−0.264 + 0.0964i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.43126 + 0.0862712i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43126 + 0.0862712i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 3 | \( 1 + (0.173 - 0.984i)T \) |
| 19 | \( 1 + (3.93 + 1.86i)T \) |
| good | 5 | \( 1 + (-0.907 - 0.761i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (0.266 + 0.460i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.939 - 1.62i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.673 + 3.82i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (1.09 - 0.397i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (5.13 - 4.30i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-3.77 - 1.37i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.979 - 1.69i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 6.88T + 37T^{2} \) |
| 41 | \( 1 + (1.56 - 8.84i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-1.85 - 1.55i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (1.91 + 0.698i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-9.93 + 8.33i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-2.51 + 0.916i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (8.69 - 7.29i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-10.4 - 3.82i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-4.65 - 3.90i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (0.0569 - 0.322i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (2.80 - 15.8i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (5.78 + 10.0i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.618 + 3.50i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (5.52 - 2.01i)T + (74.3 - 62.3i)T^{2} \) |
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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
−13.56586548075333889377058458404, −12.69714672470079687124592432724, −11.52383891080168456279188099422, −10.39833889455360864883092127693, −9.851494751771203756244383327104, −8.173759094433562470638027197992, −6.64187444688808074501151718505, −5.47053757772606299681077394104, −4.19197745442549323355301099569, −2.62755000384875629765436946495,
2.27016040909711471604506682140, 4.27285914981551215958887180856, 5.74595489274388053617687269860, 6.63272809703913071492021040875, 7.998941167135848100276310960522, 9.132756253007927017598188191639, 10.64190956828331003239921999054, 11.84667844817261556665546188464, 12.62887431565794483931322344419, 13.62345071920132524716791761622
