L(s) = 1 | + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.900 − 0.433i)8-s + (−0.365 − 0.930i)11-s + (0.365 + 0.930i)13-s + (−0.900 + 0.433i)16-s + (−0.955 + 0.294i)17-s + (0.5 − 0.866i)19-s + (−0.955 − 0.294i)22-s + (−0.733 + 0.680i)23-s + (0.955 + 0.294i)26-s + (−0.955 + 0.294i)29-s − 31-s + (−0.222 + 0.974i)32-s + (−0.365 + 0.930i)34-s + ⋯ |
L(s) = 1 | + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.900 − 0.433i)8-s + (−0.365 − 0.930i)11-s + (0.365 + 0.930i)13-s + (−0.900 + 0.433i)16-s + (−0.955 + 0.294i)17-s + (0.5 − 0.866i)19-s + (−0.955 − 0.294i)22-s + (−0.733 + 0.680i)23-s + (0.955 + 0.294i)26-s + (−0.955 + 0.294i)29-s − 31-s + (−0.222 + 0.974i)32-s + (−0.365 + 0.930i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0889 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0889 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1342378984 + 0.1227866455i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1342378984 + 0.1227866455i\) |
\(L(1)\) |
\(\approx\) |
\(0.9149171752 - 0.4850138945i\) |
\(L(1)\) |
\(\approx\) |
\(0.9149171752 - 0.4850138945i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.623 - 0.781i)T \) |
| 11 | \( 1 + (-0.365 - 0.930i)T \) |
| 13 | \( 1 + (0.365 + 0.930i)T \) |
| 17 | \( 1 + (-0.955 + 0.294i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.733 + 0.680i)T \) |
| 29 | \( 1 + (-0.955 + 0.294i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (0.733 + 0.680i)T \) |
| 41 | \( 1 + (0.826 - 0.563i)T \) |
| 43 | \( 1 + (-0.826 - 0.563i)T \) |
| 47 | \( 1 + (-0.623 + 0.781i)T \) |
| 53 | \( 1 + (-0.733 + 0.680i)T \) |
| 59 | \( 1 + (-0.900 + 0.433i)T \) |
| 61 | \( 1 + (0.222 - 0.974i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (0.222 + 0.974i)T \) |
| 73 | \( 1 + (0.365 - 0.930i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (-0.365 + 0.930i)T \) |
| 89 | \( 1 + (-0.988 + 0.149i)T \) |
| 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
−19.877368475775753370906444406071, −18.272661681915464519948267615693, −18.2119526517808154630299851595, −17.39082461695303640278745472579, −16.386752223176334814370719660433, −16.030058388165338091736970427479, −15.02686850370933740211719920549, −14.75956154460134155489767810864, −13.76144670528151653758658371903, −12.99343945552478578371168663288, −12.60523514874056560032566352604, −11.67672580956587182678961058140, −10.85713145221949436861325474179, −9.85644276525880318260973314785, −9.12010820906306611106182715050, −8.05863600609565234563094534266, −7.686504230126611995821105739737, −6.76992311380746235949632310704, −5.96808321335057472984890058903, −5.27249852814955219807316844389, −4.439594501368764186810565026893, −3.697136107033862319400867672200, −2.748281504632777202329309176849, −1.8244904805776564063160735668, −0.045334743563750794970517016339,
1.31838043046875968061223805564, 2.12644792899319310360339105675, 3.0759358302242208742195241336, 3.83454831200703043674695563710, 4.604429768095385190666157055333, 5.508403562040920974366193197135, 6.18691099772742601096680377614, 7.0131136304949644175802625725, 8.136306827445918918210399128734, 9.158286683577866569429041537670, 9.46749392649827550138629175402, 10.75027656744568327444969892507, 11.12064786357049408104738784530, 11.73371237900451124203183893126, 12.696091768673366807244850661572, 13.45369762752949240965098388500, 13.81268033162807134318081168, 14.67266144605780436162759832717, 15.541377794767887194002087780396, 16.08123494497429345245827341710, 17.02752732816332624095547519777, 18.10031491615206840552998445115, 18.51060582488623514411108592943, 19.36913004799452812316363612020, 19.92174358506036285116105083482