L(s) = 1 | + (0.0285 − 0.999i)3-s + (−0.998 + 0.0570i)5-s + (0.897 − 0.441i)7-s + (−0.998 − 0.0570i)9-s + (−0.696 + 0.717i)13-s + (0.0285 + 0.999i)15-s + (−0.974 − 0.226i)17-s + (0.516 + 0.856i)19-s + (−0.415 − 0.909i)21-s + (0.993 − 0.113i)25-s + (−0.0855 + 0.996i)27-s + (−0.516 + 0.856i)29-s + (−0.610 − 0.791i)31-s + (−0.870 + 0.491i)35-s + (0.774 + 0.633i)37-s + ⋯ |
L(s) = 1 | + (0.0285 − 0.999i)3-s + (−0.998 + 0.0570i)5-s + (0.897 − 0.441i)7-s + (−0.998 − 0.0570i)9-s + (−0.696 + 0.717i)13-s + (0.0285 + 0.999i)15-s + (−0.974 − 0.226i)17-s + (0.516 + 0.856i)19-s + (−0.415 − 0.909i)21-s + (0.993 − 0.113i)25-s + (−0.0855 + 0.996i)27-s + (−0.516 + 0.856i)29-s + (−0.610 − 0.791i)31-s + (−0.870 + 0.491i)35-s + (0.774 + 0.633i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5679528601 + 0.3169799643i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5679528601 + 0.3169799643i\) |
\(L(1)\) |
\(\approx\) |
\(0.7775784990 - 0.1432817708i\) |
\(L(1)\) |
\(\approx\) |
\(0.7775784990 - 0.1432817708i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (0.0285 - 0.999i)T \) |
| 5 | \( 1 + (-0.998 + 0.0570i)T \) |
| 7 | \( 1 + (0.897 - 0.441i)T \) |
| 13 | \( 1 + (-0.696 + 0.717i)T \) |
| 17 | \( 1 + (-0.974 - 0.226i)T \) |
| 19 | \( 1 + (0.516 + 0.856i)T \) |
| 29 | \( 1 + (-0.516 + 0.856i)T \) |
| 31 | \( 1 + (-0.610 - 0.791i)T \) |
| 37 | \( 1 + (0.774 + 0.633i)T \) |
| 41 | \( 1 + (-0.774 + 0.633i)T \) |
| 43 | \( 1 + (-0.959 + 0.281i)T \) |
| 47 | \( 1 + (-0.309 + 0.951i)T \) |
| 53 | \( 1 + (-0.466 + 0.884i)T \) |
| 59 | \( 1 + (0.985 - 0.170i)T \) |
| 61 | \( 1 + (-0.941 + 0.336i)T \) |
| 67 | \( 1 + (-0.415 - 0.909i)T \) |
| 71 | \( 1 + (0.870 + 0.491i)T \) |
| 73 | \( 1 + (0.921 + 0.389i)T \) |
| 79 | \( 1 + (0.696 - 0.717i)T \) |
| 83 | \( 1 + (-0.362 + 0.931i)T \) |
| 89 | \( 1 + (-0.959 + 0.281i)T \) |
| 97 | \( 1 + (-0.362 - 0.931i)T \) |
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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
−21.62083016688548998429482653293, −20.62564850561120725392097404116, −20.014959567019419187416449603729, −19.47221063631208863077485034006, −18.24302383128845961364725814031, −17.54996793873510733993545251834, −16.706864724951630734471747712662, −15.790290654142297260446559757053, −15.15554232093802290497161685, −14.83445012845388128918522649133, −13.72392767459402077752287794238, −12.59207346187575684882408485517, −11.62823291221986317824309896600, −11.2065884896887437434682742228, −10.36956761941209898952628128077, −9.30125672444971718608080598023, −8.54797744166340974431137975836, −7.88325520393283553431746100769, −6.879594641741996444143061508415, −5.43923430895064086264328170260, −4.90722381932839035944324129038, −4.0796574578943914814687667676, −3.14205351555762213489558669140, −2.12897951333590969256524476172, −0.29271243520177475455480721222,
1.18255579221085139328174930805, 2.09968184344243366742722075925, 3.26466802055455315859176926832, 4.33805993952917294983537750919, 5.13602411721651583976466364097, 6.43552374900371638817894093990, 7.23253057613273362944043935224, 7.807616450730756187494906418007, 8.50296254036938617816119894622, 9.54240892859831779509103542614, 10.94690139874867054821137500974, 11.44182525879911197857351636605, 12.10076171728696405664283703429, 12.961536810111617356775890830332, 13.89822490876043963498200586824, 14.57955646919588171576725784822, 15.23967196880786461863391050649, 16.55561451530043355758644700502, 16.9502140407552693494301600039, 18.18131155677162779266663773395, 18.45186135720143004242907627193, 19.48326678171336774189183540190, 20.0910764875660211167962084664, 20.63687284828148599676069551041, 21.920087357257609197920761381223