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
−19.92174358506036285116105083482, −19.36913004799452812316363612020, −18.51060582488623514411108592943, −18.10031491615206840552998445115, −17.02752732816332624095547519777, −16.08123494497429345245827341710, −15.541377794767887194002087780396, −14.67266144605780436162759832717, −13.81268033162807134318081168, −13.45369762752949240965098388500, −12.696091768673366807244850661572, −11.73371237900451124203183893126, −11.12064786357049408104738784530, −10.75027656744568327444969892507, −9.46749392649827550138629175402, −9.158286683577866569429041537670, −8.136306827445918918210399128734, −7.0131136304949644175802625725, −6.18691099772742601096680377614, −5.508403562040920974366193197135, −4.604429768095385190666157055333, −3.83454831200703043674695563710, −3.0759358302242208742195241336, −2.12644792899319310360339105675, −1.31838043046875968061223805564,
0.045334743563750794970517016339, 1.8244904805776564063160735668, 2.748281504632777202329309176849, 3.697136107033862319400867672200, 4.439594501368764186810565026893, 5.27249852814955219807316844389, 5.96808321335057472984890058903, 6.76992311380746235949632310704, 7.686504230126611995821105739737, 8.05863600609565234563094534266, 9.12010820906306611106182715050, 9.85644276525880318260973314785, 10.85713145221949436861325474179, 11.67672580956587182678961058140, 12.60523514874056560032566352604, 12.99343945552478578371168663288, 13.76144670528151653758658371903, 14.75956154460134155489767810864, 15.02686850370933740211719920549, 16.030058388165338091736970427479, 16.386752223176334814370719660433, 17.39082461695303640278745472579, 18.2119526517808154630299851595, 18.272661681915464519948267615693, 19.877368475775753370906444406071