L(s) = 1 | + (1.90 + 1.38i)2-s + (0.309 − 0.951i)3-s + (1.09 + 3.37i)4-s + (−0.809 + 0.587i)5-s + (1.90 − 1.38i)6-s + (0.0598 + 0.184i)7-s + (−1.12 + 3.47i)8-s + (−0.809 − 0.587i)9-s − 2.35·10-s + (−1.96 − 2.67i)11-s + 3.54·12-s + (−0.787 − 0.572i)13-s + (−0.140 + 0.433i)14-s + (0.309 + 0.951i)15-s + (−1.21 + 0.880i)16-s + (2.16 − 1.57i)17-s + ⋯ |
L(s) = 1 | + (1.34 + 0.979i)2-s + (0.178 − 0.549i)3-s + (0.548 + 1.68i)4-s + (−0.361 + 0.262i)5-s + (0.778 − 0.565i)6-s + (0.0226 + 0.0695i)7-s + (−0.398 + 1.22i)8-s + (−0.269 − 0.195i)9-s − 0.744·10-s + (−0.591 − 0.806i)11-s + 1.02·12-s + (−0.218 − 0.158i)13-s + (−0.0376 + 0.115i)14-s + (0.0797 + 0.245i)15-s + (−0.303 + 0.220i)16-s + (0.525 − 0.381i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.87087 + 0.987093i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.87087 + 0.987093i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.309 + 0.951i)T \) |
| 5 | \( 1 + (0.809 - 0.587i)T \) |
| 11 | \( 1 + (1.96 + 2.67i)T \) |
good | 2 | \( 1 + (-1.90 - 1.38i)T + (0.618 + 1.90i)T^{2} \) |
| 7 | \( 1 + (-0.0598 - 0.184i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (0.787 + 0.572i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-2.16 + 1.57i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (1.71 - 5.27i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 4.80T + 23T^{2} \) |
| 29 | \( 1 + (3.12 + 9.62i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.02 - 1.47i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.76 - 5.43i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (2.55 - 7.87i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 5.11T + 43T^{2} \) |
| 47 | \( 1 + (3.35 - 10.3i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-7.51 - 5.45i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (3.46 + 10.6i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (0.975 - 0.708i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 3.25T + 67T^{2} \) |
| 71 | \( 1 + (4.84 - 3.52i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.02 - 3.14i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (8.21 + 5.96i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (6.72 - 4.88i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 - 7.34T + 89T^{2} \) |
| 97 | \( 1 + (-12.8 - 9.31i)T + (29.9 + 92.2i)T^{2} \) |
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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
−13.24108365196744972660114819313, −12.30351192338643844604368302625, −11.53301337291483997580405676953, −10.01832568255525974710467851836, −8.115921650028872749056516625609, −7.72360053722438733188700399655, −6.36060329366392110630266583112, −5.63985621624884816023719803835, −4.15993027818074448142367528828, −2.90990595033121497270647484969,
2.27465340670819353207486826212, 3.69985425966542054458225769682, 4.65309934059947441333179003513, 5.58198562981746525203180816587, 7.30374377570967850289567027925, 8.822285188614603187348443268895, 10.13582852507579843179125847997, 10.82189718849627346591317122232, 11.89833356208864055005056292025, 12.62840861604668324253185950744