L(s) = 1 | + (0.5 + 0.866i)2-s + (0.724 − 1.57i)3-s + (−0.499 + 0.866i)4-s + (1.72 − 0.158i)6-s + (−2.22 − 3.85i)7-s − 0.999·8-s + (−1.94 − 2.28i)9-s + (−0.724 − 1.25i)11-s + (1 + 1.41i)12-s + (1.22 − 2.12i)13-s + (2.22 − 3.85i)14-s + (−0.5 − 0.866i)16-s + 3.89·17-s + (0.999 − 2.82i)18-s − 0.550·19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.418 − 0.908i)3-s + (−0.249 + 0.433i)4-s + (0.704 − 0.0648i)6-s + (−0.840 − 1.45i)7-s − 0.353·8-s + (−0.649 − 0.760i)9-s + (−0.218 − 0.378i)11-s + (0.288 + 0.408i)12-s + (0.339 − 0.588i)13-s + (0.594 − 1.02i)14-s + (−0.125 − 0.216i)16-s + 0.945·17-s + (0.235 − 0.666i)18-s − 0.126·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.26687 - 0.878392i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26687 - 0.878392i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.724 + 1.57i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (2.22 + 3.85i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.724 + 1.25i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.22 + 2.12i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 3.89T + 17T^{2} \) |
| 19 | \( 1 + 0.550T + 19T^{2} \) |
| 23 | \( 1 + (-1.44 + 2.51i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.22 - 5.58i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.72 + 6.45i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.224 + 0.389i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 8.44T + 53T^{2} \) |
| 59 | \( 1 + (-5.62 + 9.74i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.224 - 0.389i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.72 - 8.18i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2.44T + 71T^{2} \) |
| 73 | \( 1 + 4.79T + 73T^{2} \) |
| 79 | \( 1 + (-3.67 - 6.36i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2 - 3.46i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + (6.5 + 11.2i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.86392540245718148631061818982, −10.01578758204976328665524753827, −8.808316958454469341255902764028, −7.942057358324840163283754998058, −7.12908128955509097138178391139, −6.50661387261909198925683371210, −5.40260925178846966322644000131, −3.81502647006999228312765959573, −3.03791970702449874563586400003, −0.842468874971710756379977911906,
2.29049521467912837159803230977, 3.17759902250341638370801705926, 4.29490079782177089485085631554, 5.45079880575058505738232033927, 6.18255658830225352018272243460, 7.87669525043433559340005638348, 8.973839141647624472070439345460, 9.538453262673444440997289127460, 10.19242975926719061983461164918, 11.39417710932767387742854657225