| L(s) = 1 | + (−0.953 + 0.202i)2-s + (−1.60 + 0.660i)3-s + (−0.959 + 0.427i)4-s + (−0.978 − 0.207i)5-s + (1.39 − 0.953i)6-s + (0.137 − 1.30i)7-s + (2.40 − 1.74i)8-s + (2.12 − 2.11i)9-s + 0.974·10-s + (−2.78 + 1.80i)11-s + (1.25 − 1.31i)12-s + (−2.12 − 2.36i)13-s + (0.133 + 1.27i)14-s + (1.70 − 0.313i)15-s + (−0.532 + 0.591i)16-s + (0.549 + 1.69i)17-s + ⋯ |
| L(s) = 1 | + (−0.673 + 0.143i)2-s + (−0.924 + 0.381i)3-s + (−0.479 + 0.213i)4-s + (−0.437 − 0.0929i)5-s + (0.568 − 0.389i)6-s + (0.0519 − 0.494i)7-s + (0.850 − 0.617i)8-s + (0.709 − 0.704i)9-s + 0.308·10-s + (−0.838 + 0.544i)11-s + (0.362 − 0.380i)12-s + (−0.590 − 0.655i)13-s + (0.0358 + 0.340i)14-s + (0.439 − 0.0808i)15-s + (−0.133 + 0.147i)16-s + (0.133 + 0.410i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.415983 + 0.213991i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.415983 + 0.213991i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.60 - 0.660i)T \) |
| 5 | \( 1 + (0.978 + 0.207i)T \) |
| 11 | \( 1 + (2.78 - 1.80i)T \) |
| good | 2 | \( 1 + (0.953 - 0.202i)T + (1.82 - 0.813i)T^{2} \) |
| 7 | \( 1 + (-0.137 + 1.30i)T + (-6.84 - 1.45i)T^{2} \) |
| 13 | \( 1 + (2.12 + 2.36i)T + (-1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.549 - 1.69i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-3.04 + 2.21i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.45 + 2.52i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.940 - 8.94i)T + (-28.3 - 6.02i)T^{2} \) |
| 31 | \( 1 + (-3.97 - 4.41i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-7.60 - 5.52i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.19 - 11.3i)T + (-40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (4.49 + 7.78i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.311 - 0.138i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-3.57 + 11.0i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (8.72 - 3.88i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-3.31 + 3.67i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-2.69 + 4.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.32 - 7.16i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-4.67 - 3.39i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-4.06 + 0.864i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-3.19 + 3.54i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 + (-10.6 + 2.25i)T + (88.6 - 39.4i)T^{2} \) |
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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
−10.74195141127969555081580235044, −10.23435014392311127277479525423, −9.482131102804576623600377470665, −8.342158074005047090265691849185, −7.50553716570054438451371867600, −6.74768902767245356441483981449, −5.11391737631092576003121227638, −4.66685624752743249600750115283, −3.34664205816609222100066129290, −0.864866041511061009734289066424,
0.60916407981551615905618983295, 2.30647841218270709943052422837, 4.25537521774631953954808952861, 5.26203631540901490911387956427, 6.02830683300733785145300024563, 7.52858270886449049755117256572, 7.903859229398212722809463810512, 9.196712972603978649303328474417, 9.920380054773854932624864150709, 10.84953963909434726212265035801
