L(s) = 1 | + (0.965 + 0.258i)2-s + (0.258 + 0.965i)3-s + (0.866 + 0.499i)4-s + i·6-s + (−2.63 + 0.189i)7-s + (0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s + (−0.5 + 0.866i)11-s + (−0.258 + 0.965i)12-s + (−1.60 + 1.60i)13-s + (−2.59 − 0.5i)14-s + (0.500 + 0.866i)16-s + (−0.517 + 0.138i)17-s + (−0.965 + 0.258i)18-s + (2.86 + 4.96i)19-s + ⋯ |
L(s) = 1 | + (0.683 + 0.183i)2-s + (0.149 + 0.557i)3-s + (0.433 + 0.249i)4-s + 0.408i·6-s + (−0.997 + 0.0716i)7-s + (0.249 + 0.249i)8-s + (−0.288 + 0.166i)9-s + (−0.150 + 0.261i)11-s + (−0.0747 + 0.278i)12-s + (−0.444 + 0.444i)13-s + (−0.694 − 0.133i)14-s + (0.125 + 0.216i)16-s + (−0.125 + 0.0336i)17-s + (−0.227 + 0.0610i)18-s + (0.657 + 1.13i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 - 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.683 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.761038391\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.761038391\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 - 0.258i)T \) |
| 3 | \( 1 + (-0.258 - 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.63 - 0.189i)T \) |
good | 11 | \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.60 - 1.60i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.517 - 0.138i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-2.86 - 4.96i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.05 - 7.65i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 + (0.464 + 0.267i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (9.58 + 2.56i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 5.73iT - 41T^{2} \) |
| 43 | \( 1 + (-3.48 - 3.48i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.38 + 8.88i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.965 + 0.258i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-2.26 + 3.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.92 + 2.26i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.517 - 1.93i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 4.92T + 71T^{2} \) |
| 73 | \( 1 + (-1.03 - 3.86i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-14.6 + 8.46i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7.34 - 7.34i)T - 83iT^{2} \) |
| 89 | \( 1 + (3.46 + 6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.72 - 7.72i)T + 97iT^{2} \) |
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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
−10.10842688453413665708912541464, −9.536834671509127019595628568084, −8.647196860076006033309166325467, −7.49439065300050429670987517987, −6.86027211901637648758829043146, −5.71952356522377526677149494360, −5.15767921178089361936112170466, −3.83257202672997005964897031956, −3.36932458917577782962607224021, −2.00908150847754060703881411237,
0.58562055757380370034116585199, 2.40872892476992567989264722209, 3.07498675517438374035368571166, 4.24812951471260870745557838280, 5.32456713153030474728896601208, 6.27582963237795522767678454783, 6.88814802655297114364174996110, 7.78155734715097976999867351782, 8.789220783998389031882032623562, 9.715275643534491130409951719184