L(s) = 1 | + (0.766 + 0.642i)2-s + (−0.646 + 0.542i)3-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)5-s − 0.843·6-s + (0.282 + 0.102i)7-s + (−0.500 + 0.866i)8-s + (−0.397 + 2.25i)9-s + (−0.5 − 0.866i)10-s + (−3.09 + 5.35i)11-s + (−0.646 − 0.542i)12-s + (−0.104 − 0.593i)13-s + (0.150 + 0.260i)14-s + (0.792 − 0.288i)15-s + (−0.939 + 0.342i)16-s + (−0.274 + 1.55i)17-s + ⋯ |
L(s) = 1 | + (0.541 + 0.454i)2-s + (−0.373 + 0.313i)3-s + (0.0868 + 0.492i)4-s + (−0.420 − 0.152i)5-s − 0.344·6-s + (0.106 + 0.0388i)7-s + (−0.176 + 0.306i)8-s + (−0.132 + 0.751i)9-s + (−0.158 − 0.273i)10-s + (−0.931 + 1.61i)11-s + (−0.186 − 0.156i)12-s + (−0.0290 − 0.164i)13-s + (0.0401 + 0.0694i)14-s + (0.204 − 0.0744i)15-s + (−0.234 + 0.0855i)16-s + (−0.0665 + 0.377i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.651 - 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.515395 + 1.12158i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.515395 + 1.12158i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 5 | \( 1 + (0.939 + 0.342i)T \) |
| 37 | \( 1 + (-5.42 + 2.74i)T \) |
good | 3 | \( 1 + (0.646 - 0.542i)T + (0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (-0.282 - 0.102i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (3.09 - 5.35i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.104 + 0.593i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.274 - 1.55i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (-3.68 + 3.08i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (-3.32 - 5.75i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.0493 - 0.0854i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 9.46T + 31T^{2} \) |
| 41 | \( 1 + (1.46 + 8.30i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 - 5.55T + 43T^{2} \) |
| 47 | \( 1 + (-5.91 - 10.2i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-12.8 + 4.67i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-10.9 + 3.99i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.63 - 9.26i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.75 - 1.36i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (6.27 - 5.26i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + 4.14T + 73T^{2} \) |
| 79 | \( 1 + (-6.38 - 2.32i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-1.46 + 8.28i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-5.34 + 1.94i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (5.32 + 9.23i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.72034368713217257688563512033, −10.96332212724695811211319925186, −9.997422551434000539381435928344, −8.900618753448571667781598854624, −7.45546338870212956669625181474, −7.39015543755081615388653224409, −5.53135671272530963189853856109, −5.04165650181581730654118454930, −3.96429547386679690307913856111, −2.35841535030215127547625688334,
0.73898235793506164080505871052, 2.82761269349839925751937545859, 3.78208562158474579406652551670, 5.25467663417446560403453324487, 6.03845787552011003595392771348, 7.12373351135358170124798664478, 8.265075669054989208802623200159, 9.282814305207035389162202519362, 10.54101398948912199734706154489, 11.22732607513294963347052209356