L(s) = 1 | + (0.866 + 0.5i)3-s + (0.5 + 0.866i)7-s + (0.5 + 0.866i)9-s + (0.866 + 0.5i)11-s + (0.866 − 0.5i)17-s + (−0.866 + 0.5i)19-s + i·21-s + (−0.866 − 0.5i)23-s − i·27-s + (0.5 − 0.866i)29-s − i·31-s + (0.5 + 0.866i)33-s + (−0.5 + 0.866i)37-s + (0.866 + 0.5i)41-s + (0.866 − 0.5i)43-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)3-s + (0.5 + 0.866i)7-s + (0.5 + 0.866i)9-s + (0.866 + 0.5i)11-s + (0.866 − 0.5i)17-s + (−0.866 + 0.5i)19-s + i·21-s + (−0.866 − 0.5i)23-s − i·27-s + (0.5 − 0.866i)29-s − i·31-s + (0.5 + 0.866i)33-s + (−0.5 + 0.866i)37-s + (0.866 + 0.5i)41-s + (0.866 − 0.5i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0557 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0557 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.082104629 + 1.969115855i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.082104629 + 1.969115855i\) |
\(L(1)\) |
\(\approx\) |
\(1.489612934 + 0.5928177049i\) |
\(L(1)\) |
\(\approx\) |
\(1.489612934 + 0.5928177049i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 + (0.866 + 0.5i)T \) |
| 17 | \( 1 + (0.866 - 0.5i)T \) |
| 19 | \( 1 + (-0.866 + 0.5i)T \) |
| 23 | \( 1 + (-0.866 - 0.5i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 - iT \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (0.866 + 0.5i)T \) |
| 43 | \( 1 + (0.866 - 0.5i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + iT \) |
| 59 | \( 1 + (0.866 - 0.5i)T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.866 - 0.5i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (0.866 + 0.5i)T \) |
| 97 | \( 1 + (0.5 + 0.866i)T \) |
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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
−25.609928525496249969760490508028, −24.338696595755289127782796854452, −23.95525527190320710567924544599, −22.87002273991379015596136975158, −21.51421978894782471066454841388, −20.83836683668052808727683235323, −19.66792852571526214571944127928, −19.355088536170507747736171539515, −18.036791970359894651265039311198, −17.22553925313270079782667177460, −16.14904249724124829900723892958, −14.75815783342555821634010002793, −14.25552010622772509002544400720, −13.34400269550641538031773964569, −12.33062578430841475029840590081, −11.190713688104321188599274150048, −10.03872232815100834126117685171, −8.91170966368642737967016626773, −8.0028484320768963812240374991, −7.11088036803295227623608167762, −6.00854940772366059449467329106, −4.276456201668533361753918077178, −3.46099137119749701294810228447, −1.945944693560030199286967668419, −0.84649959551242823592445366488,
1.620829025734895824789997321, 2.69495797952331663077130901780, 3.99146479632547844758146398677, 4.97075667958247345723760919903, 6.31909051787402016574625971542, 7.75704344684916374978258363903, 8.57836567182475766041940261705, 9.49942807461727782656789491916, 10.40973544167008436044498981509, 11.77149336583541019828886085822, 12.58851817695515701670273623729, 14.06035159168539424278638175408, 14.57793898768292433121263961614, 15.47707507582957527131861744679, 16.41860063250843570475769480350, 17.5793800641801867498970742458, 18.70502914723781652937307914367, 19.451167818614300717044295689919, 20.508937689041147240214038765296, 21.2219261080870796770258535115, 22.040909057816395834514178876772, 23.01917832168750042161800621502, 24.3859201765798405849477434010, 25.12011073536699745688011991649, 25.6888165900975786166420230788