L(s) = 1 | + (1.29 − 1.49i)2-s + (0.749 − 0.481i)3-s + (−0.274 − 1.90i)4-s + (0.415 + 0.909i)5-s + (0.251 − 1.74i)6-s + (−3.44 + 1.01i)7-s + (0.118 + 0.0762i)8-s + (−0.916 + 2.00i)9-s + (1.90 + 0.558i)10-s + (−2.66 − 3.07i)11-s + (−1.12 − 1.29i)12-s + (0.453 + 0.133i)13-s + (−2.95 + 6.47i)14-s + (0.749 + 0.481i)15-s + (3.96 − 1.16i)16-s + (−0.293 + 2.04i)17-s + ⋯ |
L(s) = 1 | + (0.917 − 1.05i)2-s + (0.432 − 0.278i)3-s + (−0.137 − 0.954i)4-s + (0.185 + 0.406i)5-s + (0.102 − 0.713i)6-s + (−1.30 + 0.382i)7-s + (0.0419 + 0.0269i)8-s + (−0.305 + 0.668i)9-s + (0.601 + 0.176i)10-s + (−0.803 − 0.927i)11-s + (−0.324 − 0.374i)12-s + (0.125 + 0.0369i)13-s + (−0.789 + 1.72i)14-s + (0.193 + 0.124i)15-s + (0.992 − 0.291i)16-s + (−0.0712 + 0.495i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.35597 - 0.932119i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.35597 - 0.932119i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (-3.46 - 3.31i)T \) |
good | 2 | \( 1 + (-1.29 + 1.49i)T + (-0.284 - 1.97i)T^{2} \) |
| 3 | \( 1 + (-0.749 + 0.481i)T + (1.24 - 2.72i)T^{2} \) |
| 7 | \( 1 + (3.44 - 1.01i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (2.66 + 3.07i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.453 - 0.133i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.293 - 2.04i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (1.10 + 7.67i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.637 + 4.43i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-4.14 - 2.66i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-0.140 + 0.306i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-1.21 - 2.65i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-2.28 + 1.46i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 9.09T + 47T^{2} \) |
| 53 | \( 1 + (4.31 - 1.26i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (14.0 + 4.12i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-3.64 - 2.34i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (0.224 - 0.259i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-6.18 + 7.13i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (0.367 + 2.55i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-16.5 - 4.86i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (1.07 - 2.35i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (7.82 - 5.02i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-6.27 - 13.7i)T + (-63.5 + 73.3i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.43743892938195797420042335216, −12.63783779782513450026972865206, −11.27820191448396762555793155636, −10.64864480074470671950168689443, −9.339300173730871792172397044829, −7.988128856525370930784835678138, −6.40283891959870817880546888677, −5.09023698659922611471583381210, −3.23285776136797172714924761229, −2.57043539026161584299026106975,
3.30583104773865710796504159173, 4.56673263289293307876207558746, 5.93854869949751128617649106428, 6.85910561094829664224999392537, 8.107055321707819537651008151491, 9.531030626602611767428766151310, 10.30795272941466899738504120187, 12.41565092767705846723939342951, 12.88573463111070749241308169962, 13.94265902820235978077881382247