| L(s) = 1 | + (−0.587 − 0.809i)2-s + (0.195 + 0.0635i)3-s + (−0.309 + 0.951i)4-s + (0.0419 − 2.23i)5-s + (−0.0635 − 0.195i)6-s − i·7-s + (0.951 − 0.309i)8-s + (−2.39 − 1.73i)9-s + (−1.83 + 1.28i)10-s + (0.483 − 0.351i)11-s + (−0.120 + 0.166i)12-s + (−1.59 + 2.19i)13-s + (−0.809 + 0.587i)14-s + (0.150 − 0.434i)15-s + (−0.809 − 0.587i)16-s + (1.72 − 0.559i)17-s + ⋯ |
| L(s) = 1 | + (−0.415 − 0.572i)2-s + (0.112 + 0.0366i)3-s + (−0.154 + 0.475i)4-s + (0.0187 − 0.999i)5-s + (−0.0259 − 0.0797i)6-s − 0.377i·7-s + (0.336 − 0.109i)8-s + (−0.797 − 0.579i)9-s + (−0.579 + 0.404i)10-s + (0.145 − 0.105i)11-s + (−0.0348 + 0.0479i)12-s + (−0.443 + 0.610i)13-s + (−0.216 + 0.157i)14-s + (0.0387 − 0.112i)15-s + (−0.202 − 0.146i)16-s + (0.417 − 0.135i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.297268 - 0.794039i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.297268 - 0.794039i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.587 + 0.809i)T \) |
| 5 | \( 1 + (-0.0419 + 2.23i)T \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 + (-0.195 - 0.0635i)T + (2.42 + 1.76i)T^{2} \) |
| 11 | \( 1 + (-0.483 + 0.351i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (1.59 - 2.19i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.72 + 0.559i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (2.23 + 6.88i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.516 + 0.711i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.25 + 3.85i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.36 + 4.19i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (2.27 - 3.13i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-4.93 - 3.58i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 0.789iT - 43T^{2} \) |
| 47 | \( 1 + (-0.430 - 0.139i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-9.74 - 3.16i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-1.21 - 0.883i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-9.35 + 6.79i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (3.66 - 1.19i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (0.329 - 1.01i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-4.35 - 5.99i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-1.32 + 4.07i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.811 + 0.263i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-9.67 + 7.03i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-3.27 - 1.06i)T + (78.4 + 57.0i)T^{2} \) |
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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
−11.36293142478368549793169208913, −10.05589225357650548997402180600, −9.205820861438424731227911918878, −8.655149656645857004074687702055, −7.58234442868040901877213881189, −6.31745606167091866228498477549, −4.93865883920723496996155159685, −3.92315888311136337911188905324, −2.44113946101840936394641491676, −0.65239516430987762873458528769,
2.20216226776400900562327406246, 3.55255283934624666308278293335, 5.33310211466907321867670663275, 6.06153530428061710331941117952, 7.23286606906408957608355045113, 8.012793397770523619153751360427, 8.888552488512057177843670954247, 10.15734084013654151088754425499, 10.60761958747977349254185569059, 11.72872618627350112380522614096
