L(s) = 1 | + (0.372 + 0.927i)2-s + (−0.967 + 0.251i)3-s + (−0.721 + 0.691i)4-s + (−0.450 − 0.892i)5-s + (−0.594 − 0.803i)6-s + (0.660 − 0.750i)7-s + (−0.911 − 0.411i)8-s + (0.873 − 0.487i)9-s + (0.660 − 0.750i)10-s + (−0.721 − 0.691i)11-s + (0.524 − 0.851i)12-s + (0.985 + 0.169i)13-s + (0.942 + 0.333i)14-s + (0.660 + 0.750i)15-s + (0.0424 − 0.999i)16-s + (−0.967 + 0.251i)17-s + ⋯ |
L(s) = 1 | + (0.372 + 0.927i)2-s + (−0.967 + 0.251i)3-s + (−0.721 + 0.691i)4-s + (−0.450 − 0.892i)5-s + (−0.594 − 0.803i)6-s + (0.660 − 0.750i)7-s + (−0.911 − 0.411i)8-s + (0.873 − 0.487i)9-s + (0.660 − 0.750i)10-s + (−0.721 − 0.691i)11-s + (0.524 − 0.851i)12-s + (0.985 + 0.169i)13-s + (0.942 + 0.333i)14-s + (0.660 + 0.750i)15-s + (0.0424 − 0.999i)16-s + (−0.967 + 0.251i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7832083750 - 0.04532660565i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7832083750 - 0.04532660565i\) |
\(L(1)\) |
\(\approx\) |
\(0.8180217651 + 0.1639113392i\) |
\(L(1)\) |
\(\approx\) |
\(0.8180217651 + 0.1639113392i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 149 | \( 1 \) |
good | 2 | \( 1 + (0.372 + 0.927i)T \) |
| 3 | \( 1 + (-0.967 + 0.251i)T \) |
| 5 | \( 1 + (-0.450 - 0.892i)T \) |
| 7 | \( 1 + (0.660 - 0.750i)T \) |
| 11 | \( 1 + (-0.721 - 0.691i)T \) |
| 13 | \( 1 + (0.985 + 0.169i)T \) |
| 17 | \( 1 + (-0.967 + 0.251i)T \) |
| 19 | \( 1 + (0.778 - 0.628i)T \) |
| 23 | \( 1 + (0.985 - 0.169i)T \) |
| 29 | \( 1 + (-0.292 - 0.956i)T \) |
| 31 | \( 1 + (0.985 - 0.169i)T \) |
| 37 | \( 1 + (-0.721 - 0.691i)T \) |
| 41 | \( 1 + (0.0424 - 0.999i)T \) |
| 43 | \( 1 + (-0.828 - 0.559i)T \) |
| 47 | \( 1 + (0.778 + 0.628i)T \) |
| 53 | \( 1 + (-0.828 + 0.559i)T \) |
| 59 | \( 1 + (0.210 - 0.977i)T \) |
| 61 | \( 1 + (0.372 + 0.927i)T \) |
| 67 | \( 1 + (0.942 - 0.333i)T \) |
| 71 | \( 1 + (-0.450 - 0.892i)T \) |
| 73 | \( 1 + (-0.127 + 0.991i)T \) |
| 79 | \( 1 + (-0.127 - 0.991i)T \) |
| 83 | \( 1 + (-0.127 + 0.991i)T \) |
| 89 | \( 1 + (-0.996 - 0.0848i)T \) |
| 97 | \( 1 + (-0.828 + 0.559i)T \) |
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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
−28.30578806964099897295265006797, −27.50162692821013857341638602077, −26.59247797767421034463571858182, −24.95712284604391026759468628114, −23.753434386984102012036993010149, −23.02777905386470523825789003874, −22.32543326941636746774849385657, −21.3798966101899737505851741081, −20.369247435327301713015202255026, −18.906020420154850901144128024093, −18.28117581865029011581035193069, −17.72115214772090467884611598657, −15.77472178087257617836528524420, −15.02268227140115454470700552424, −13.65042404261054099879939465251, −12.56102128042703277160069920953, −11.534783634093737711909079671950, −11.009010479378171793458729893453, −9.97595745740370272965701181145, −8.31306549528364924927014551165, −6.82541071504795749143901660182, −5.54847951993244566837673046934, −4.5788735644679691292327596618, −2.99887092710881795354479876619, −1.607517062922305534717207915552,
0.7464069209382251149287256938, 3.81789472996379958549890470159, 4.72295197229134582269638367561, 5.59237600604975534693613626753, 6.899398592427760094211816421466, 8.05503434831298717454264664055, 9.095699170932761374462128526984, 10.81192857758172787409140849434, 11.70878844909284821258095238246, 13.04573814155314736057692574890, 13.70729618361030501185664361159, 15.49156669780762654377853156750, 15.91757595526761301414065951095, 17.01284211975645356723542040561, 17.59421541291729918990299193461, 18.81572079213723028672788585866, 20.65171660794380086817007531956, 21.234077962820129344243412180, 22.51889836290484815675451171358, 23.46961751269729312516761564179, 23.99267750430401018973543473205, 24.712554242618306624315122687908, 26.46894058101870836148805077275, 26.89077542686342092417071237089, 28.03411845692154407696923588802