L(s) = 1 | + (0.309 + 0.951i)3-s − 7-s + (−0.809 + 0.587i)9-s + (−0.809 − 0.587i)11-s + (0.809 − 0.587i)13-s + (0.309 − 0.951i)17-s + (0.309 − 0.951i)19-s + (−0.309 − 0.951i)21-s + (0.809 + 0.587i)23-s + (−0.809 − 0.587i)27-s + (−0.309 − 0.951i)29-s + (−0.309 + 0.951i)31-s + (0.309 − 0.951i)33-s + (0.809 − 0.587i)37-s + (0.809 + 0.587i)39-s + ⋯ |
L(s) = 1 | + (0.309 + 0.951i)3-s − 7-s + (−0.809 + 0.587i)9-s + (−0.809 − 0.587i)11-s + (0.809 − 0.587i)13-s + (0.309 − 0.951i)17-s + (0.309 − 0.951i)19-s + (−0.309 − 0.951i)21-s + (0.809 + 0.587i)23-s + (−0.809 − 0.587i)27-s + (−0.309 − 0.951i)29-s + (−0.309 + 0.951i)31-s + (0.309 − 0.951i)33-s + (0.809 − 0.587i)37-s + (0.809 + 0.587i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.010566811 - 0.5555638060i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.010566811 - 0.5555638060i\) |
\(L(1)\) |
\(\approx\) |
\(0.9515917494 + 0.07769500621i\) |
\(L(1)\) |
\(\approx\) |
\(0.9515917494 + 0.07769500621i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.309 + 0.951i)T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + (-0.809 - 0.587i)T \) |
| 13 | \( 1 + (0.809 - 0.587i)T \) |
| 17 | \( 1 + (0.309 - 0.951i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
| 23 | \( 1 + (0.809 + 0.587i)T \) |
| 29 | \( 1 + (-0.309 - 0.951i)T \) |
| 31 | \( 1 + (-0.309 + 0.951i)T \) |
| 37 | \( 1 + (0.809 - 0.587i)T \) |
| 41 | \( 1 + (-0.809 + 0.587i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.309 - 0.951i)T \) |
| 53 | \( 1 + (-0.309 - 0.951i)T \) |
| 59 | \( 1 + (-0.809 + 0.587i)T \) |
| 61 | \( 1 + (0.809 + 0.587i)T \) |
| 67 | \( 1 + (0.309 - 0.951i)T \) |
| 71 | \( 1 + (-0.309 - 0.951i)T \) |
| 73 | \( 1 + (-0.809 - 0.587i)T \) |
| 79 | \( 1 + (-0.309 - 0.951i)T \) |
| 83 | \( 1 + (0.309 - 0.951i)T \) |
| 89 | \( 1 + (-0.809 - 0.587i)T \) |
| 97 | \( 1 + (0.309 + 0.951i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−26.40754020006331543770720784453, −25.80432435185185025933993665140, −25.11305988843245210508319845518, −23.82409909086624557374879054346, −23.27954347219578669297242341145, −22.29016449351910396480688020274, −20.86052869546559381336638490592, −20.14376488519479207999598952953, −18.786042830504125335884625263767, −18.687541435365082261363757431098, −17.255879423597134277035313675084, −16.273427828882776078061345301397, −15.09354619944443923294532384744, −14.02828818258979114721733412602, −12.909530391291307235244743095023, −12.53114242924453111591028007171, −11.09568683507663318474373197594, −9.84594415403781327451624492318, −8.7181810865794962227399427155, −7.64950312219785916594533919685, −6.609954297645564908690036971735, −5.69711709100122453254212130681, −3.84191785919302116402084238772, −2.67141491377310327627469545944, −1.31873628799363932254441788048,
0.39651838074974595363975969880, 2.809264169110000621506073988252, 3.45751433916472467897829498147, 4.97876513368106918503544536736, 5.94025910760982469417560421493, 7.46136092204688491461674808927, 8.73012245151757562446286088750, 9.58290952342709146130783144394, 10.57566120306837972709365227346, 11.485220816082407740772102921185, 13.120312059098905145862901485113, 13.70993447113202602312519077954, 15.14759017729638720362907359783, 15.90455017359143604236991707705, 16.499995086244376745911884912788, 17.88187819984667958119440144068, 19.027175516219352585713928890696, 19.95408306827023235315219395649, 20.87250489691061914199410043787, 21.698689279438185228223669648947, 22.70812172877624915035733063031, 23.39839570667207158886730019318, 24.94420070859002608224565909467, 25.704541652692474066951607783882, 26.50295120049098763061238647420