L(s) = 1 | + (0.203 + 0.979i)2-s + (0.203 + 0.979i)3-s + (−0.917 + 0.398i)4-s + (0.460 − 0.887i)5-s + (−0.917 + 0.398i)6-s + (−0.917 − 0.398i)7-s + (−0.576 − 0.816i)8-s + (−0.917 + 0.398i)9-s + (0.962 + 0.269i)10-s + (0.854 − 0.519i)11-s + (−0.576 − 0.816i)12-s + (0.962 − 0.269i)13-s + (0.203 − 0.979i)14-s + (0.962 + 0.269i)15-s + (0.682 − 0.730i)16-s + (0.962 − 0.269i)17-s + ⋯ |
L(s) = 1 | + (0.203 + 0.979i)2-s + (0.203 + 0.979i)3-s + (−0.917 + 0.398i)4-s + (0.460 − 0.887i)5-s + (−0.917 + 0.398i)6-s + (−0.917 − 0.398i)7-s + (−0.576 − 0.816i)8-s + (−0.917 + 0.398i)9-s + (0.962 + 0.269i)10-s + (0.854 − 0.519i)11-s + (−0.576 − 0.816i)12-s + (0.962 − 0.269i)13-s + (0.203 − 0.979i)14-s + (0.962 + 0.269i)15-s + (0.682 − 0.730i)16-s + (0.962 − 0.269i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.158150907 + 0.9770247485i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.158150907 + 0.9770247485i\) |
\(L(1)\) |
\(\approx\) |
\(1.009482613 + 0.6554546219i\) |
\(L(1)\) |
\(\approx\) |
\(1.009482613 + 0.6554546219i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (0.203 + 0.979i)T \) |
| 3 | \( 1 + (0.203 + 0.979i)T \) |
| 5 | \( 1 + (0.460 - 0.887i)T \) |
| 7 | \( 1 + (-0.917 - 0.398i)T \) |
| 11 | \( 1 + (0.854 - 0.519i)T \) |
| 13 | \( 1 + (0.962 - 0.269i)T \) |
| 17 | \( 1 + (0.962 - 0.269i)T \) |
| 19 | \( 1 + (0.203 + 0.979i)T \) |
| 29 | \( 1 + (0.854 - 0.519i)T \) |
| 31 | \( 1 + (-0.0682 + 0.997i)T \) |
| 37 | \( 1 + (0.682 + 0.730i)T \) |
| 41 | \( 1 + (-0.990 + 0.136i)T \) |
| 43 | \( 1 + (-0.334 + 0.942i)T \) |
| 47 | \( 1 + (-0.0682 - 0.997i)T \) |
| 53 | \( 1 + (0.962 - 0.269i)T \) |
| 59 | \( 1 + (0.203 - 0.979i)T \) |
| 61 | \( 1 + (0.460 - 0.887i)T \) |
| 67 | \( 1 + (0.854 - 0.519i)T \) |
| 71 | \( 1 + (-0.990 - 0.136i)T \) |
| 73 | \( 1 + (0.854 + 0.519i)T \) |
| 79 | \( 1 + (-0.334 + 0.942i)T \) |
| 83 | \( 1 + (0.460 - 0.887i)T \) |
| 89 | \( 1 + (-0.775 + 0.631i)T \) |
| 97 | \( 1 + (-0.990 - 0.136i)T \) |
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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
−23.115520167054254478344359365161, −22.42679406682444655993894393959, −21.75198176779652984425056262452, −20.69579321212056370080127987834, −19.70373366301767241272728998994, −19.16462979253849979563947186370, −18.431025811763537613700621398352, −17.84607351512039094910963907120, −16.84187782772451952816182276579, −15.25708137397321555713210729297, −14.432287665118526664932393210992, −13.65825959031427372097913457798, −13.00973296638722309297565585703, −12.06830648043619514893074305401, −11.407505710699884573230462699634, −10.298301187297729309745859333850, −9.38903169179815387453111578955, −8.6690742422405031972122105160, −7.208936002212241490342876890421, −6.33197703459222382804851151638, −5.62766794448109032330617633615, −3.86294165269656049422852610686, −3.02585323100768837516375240725, −2.19874327450552633647380064824, −1.10336351296026361411084194863,
0.98067139977839516090036682509, 3.28644646074419202958940574690, 3.829040441887682498882913174507, 4.94997600830731210401550800486, 5.83236373686174276490331447310, 6.526749125947998277660935180444, 8.12873618738077172408980327189, 8.666188521115193117520224298178, 9.69324278141754533520336851024, 10.10709228814080919008124580912, 11.73015151338228123163987792846, 12.76568134865185202708225417408, 13.76129151437992844875058405777, 14.165804223524584533322493836512, 15.36281560837221169667221531729, 16.3360523260700631207638505252, 16.47337282321123258470011847038, 17.26955315168535204300142975002, 18.475628252731578156211975873261, 19.58767884674736135762083621758, 20.494410612948896580745529452341, 21.33999028138876915392096662003, 22.00582220204972943138896966072, 23.01616454906061951253483500463, 23.42787381242598141434881900178