| L(s) = 1 | + (−0.0713 + 0.997i)2-s + (0.977 − 0.212i)3-s + (−0.989 − 0.142i)4-s + (−1.67 + 1.48i)5-s + (0.142 + 0.989i)6-s + (3.33 − 1.81i)7-s + (0.212 − 0.977i)8-s + (0.909 − 0.415i)9-s + (−1.36 − 1.77i)10-s + (−0.824 − 0.714i)11-s + (−0.997 + 0.0713i)12-s + (1.01 − 1.85i)13-s + (1.57 + 3.45i)14-s + (−1.31 + 1.80i)15-s + (0.959 + 0.281i)16-s + (6.37 + 4.77i)17-s + ⋯ |
| L(s) = 1 | + (−0.0504 + 0.705i)2-s + (0.564 − 0.122i)3-s + (−0.494 − 0.0711i)4-s + (−0.747 + 0.664i)5-s + (0.0580 + 0.404i)6-s + (1.25 − 0.687i)7-s + (0.0751 − 0.345i)8-s + (0.303 − 0.138i)9-s + (−0.430 − 0.560i)10-s + (−0.248 − 0.215i)11-s + (−0.287 + 0.0205i)12-s + (0.280 − 0.514i)13-s + (0.421 + 0.922i)14-s + (−0.340 + 0.466i)15-s + (0.239 + 0.0704i)16-s + (1.54 + 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.56348 + 0.809112i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.56348 + 0.809112i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0713 - 0.997i)T \) |
| 3 | \( 1 + (-0.977 + 0.212i)T \) |
| 5 | \( 1 + (1.67 - 1.48i)T \) |
| 23 | \( 1 + (-4.50 - 1.64i)T \) |
| good | 7 | \( 1 + (-3.33 + 1.81i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (0.824 + 0.714i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.01 + 1.85i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-6.37 - 4.77i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (0.435 - 3.03i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (3.03 - 0.436i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-4.68 + 3.01i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (3.91 + 1.45i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (3.28 - 7.19i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-1.55 - 7.16i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-2.60 - 2.60i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.67 + 8.56i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (2.33 + 7.94i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-2.89 - 4.51i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (0.125 + 0.00898i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-4.04 - 4.67i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (2.47 + 3.31i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-14.2 + 4.16i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (0.411 - 1.10i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (1.56 + 1.00i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (2.58 + 6.93i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−10.58411125285672546586315697667, −9.754644931407209180227903902147, −8.229670265288949896305427954808, −8.073012560940361558825499355439, −7.41829129674514878237496784287, −6.30759986358203799418108517166, −5.17571759661092045707657673141, −4.04012868235076393984625290178, −3.23036204124004586463933416430, −1.32448212773361955170021037471,
1.18510770728669903547794049621, 2.50421981039815114471451452037, 3.67085232910156657745893351523, 4.84898359765890802148772701443, 5.24503377033819684674635640066, 7.21228174643095746282786883534, 7.967243887294540588212456009468, 8.797473849983798008133874393457, 9.219253754957347554360747970862, 10.43060373207527024870984308784
