L(s) = 1 | + (0.0312 + 0.999i)3-s + (0.915 − 0.401i)5-s + (−0.570 + 0.821i)7-s + (−0.998 + 0.0625i)9-s + (−0.0687 − 0.997i)11-s + (−0.580 + 0.814i)13-s + (0.429 + 0.902i)15-s + (−0.943 + 0.331i)17-s + (−0.877 + 0.479i)19-s + (−0.838 − 0.544i)21-s + (0.217 − 0.976i)23-s + (0.677 − 0.735i)25-s + (−0.0937 − 0.995i)27-s + (0.253 + 0.967i)29-s + (−0.971 + 0.235i)31-s + ⋯ |
L(s) = 1 | + (0.0312 + 0.999i)3-s + (0.915 − 0.401i)5-s + (−0.570 + 0.821i)7-s + (−0.998 + 0.0625i)9-s + (−0.0687 − 0.997i)11-s + (−0.580 + 0.814i)13-s + (0.429 + 0.902i)15-s + (−0.943 + 0.331i)17-s + (−0.877 + 0.479i)19-s + (−0.838 − 0.544i)21-s + (0.217 − 0.976i)23-s + (0.677 − 0.735i)25-s + (−0.0937 − 0.995i)27-s + (0.253 + 0.967i)29-s + (−0.971 + 0.235i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6626140735 - 0.3362800384i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6626140735 - 0.3362800384i\) |
\(L(1)\) |
\(\approx\) |
\(0.8783440164 + 0.2361102155i\) |
\(L(1)\) |
\(\approx\) |
\(0.8783440164 + 0.2361102155i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 \) |
good | 3 | \( 1 + (0.0312 + 0.999i)T \) |
| 5 | \( 1 + (0.915 - 0.401i)T \) |
| 7 | \( 1 + (-0.570 + 0.821i)T \) |
| 11 | \( 1 + (-0.0687 - 0.997i)T \) |
| 13 | \( 1 + (-0.580 + 0.814i)T \) |
| 17 | \( 1 + (-0.943 + 0.331i)T \) |
| 19 | \( 1 + (-0.877 + 0.479i)T \) |
| 23 | \( 1 + (0.217 - 0.976i)T \) |
| 29 | \( 1 + (0.253 + 0.967i)T \) |
| 31 | \( 1 + (-0.971 + 0.235i)T \) |
| 37 | \( 1 + (-0.764 - 0.644i)T \) |
| 41 | \( 1 + (0.947 + 0.319i)T \) |
| 43 | \( 1 + (0.528 + 0.848i)T \) |
| 47 | \( 1 + (0.474 - 0.880i)T \) |
| 53 | \( 1 + (-0.877 - 0.479i)T \) |
| 59 | \( 1 + (0.301 - 0.953i)T \) |
| 61 | \( 1 + (-0.994 - 0.0999i)T \) |
| 67 | \( 1 + (-0.429 + 0.902i)T \) |
| 71 | \( 1 + (0.731 + 0.682i)T \) |
| 73 | \( 1 + (0.407 - 0.913i)T \) |
| 79 | \( 1 + (0.0437 + 0.999i)T \) |
| 83 | \( 1 + (-0.980 + 0.198i)T \) |
| 89 | \( 1 + (0.999 - 0.0250i)T \) |
| 97 | \( 1 + (0.168 - 0.985i)T \) |
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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
−18.51120791138743144241972039134, −17.71241458191443846949172462421, −17.30198382475816701555212347560, −17.068917060091252037573925223111, −15.71688409306107324079869171266, −15.11430529082755563870918834294, −14.30976529884563497977539282908, −13.64863408160437483896854664708, −13.11153379369664900673079594613, −12.69644176762610061531159383211, −11.84671863924197374140827061358, −10.78352209893524261074938458740, −10.48101854939343451696613818409, −9.41586263811118864931101197118, −9.095166240976178438889767022119, −7.72770254058613636122087281817, −7.37824774150470011473344306247, −6.67051121741996210198165142541, −6.113679490507089969752628080125, −5.25434360664017551888925394862, −4.38108957722395391660135328916, −3.2769847152472961159963176488, −2.46550273680680265334004878747, −1.98216762857592230000105204076, −0.922320745859226081499902303399,
0.21598136654738256262451803372, 1.817446547519231255948180841390, 2.46976635920430015282865653492, 3.23965934404457423536881315798, 4.20014082804617102113721938270, 4.885869063787370144687816013610, 5.66749852527256772409834112804, 6.18228949755064929765179965575, 6.882569020855337235685211417055, 8.454282675440325665970465854106, 8.7196118804023286259078058713, 9.31694387393827309100450926873, 9.96052806901233689367297154110, 10.80999376293885427327679645810, 11.21532770651947719903714967658, 12.47499926937033478371425608564, 12.685474231248198069882033978161, 13.76646521670563956499169346447, 14.37481446864873365227840452600, 14.88712021724526167243870255789, 15.85161494702136399572096175502, 16.388159568339100043374836459961, 16.76461778182676363755402338679, 17.57579643967162381639640631886, 18.328577196690596304759474637240