L(s) = 1 | + (−0.173 − 0.984i)3-s + (−0.5 + 0.866i)7-s + (−0.939 + 0.342i)9-s + (−0.5 − 0.866i)11-s + (0.173 − 0.984i)13-s + (0.939 + 0.342i)17-s + (0.939 + 0.342i)21-s + (0.766 + 0.642i)23-s + (0.5 + 0.866i)27-s + (0.939 − 0.342i)29-s + (0.5 − 0.866i)31-s + (−0.766 + 0.642i)33-s + 37-s − 39-s + (0.173 + 0.984i)41-s + ⋯ |
L(s) = 1 | + (−0.173 − 0.984i)3-s + (−0.5 + 0.866i)7-s + (−0.939 + 0.342i)9-s + (−0.5 − 0.866i)11-s + (0.173 − 0.984i)13-s + (0.939 + 0.342i)17-s + (0.939 + 0.342i)21-s + (0.766 + 0.642i)23-s + (0.5 + 0.866i)27-s + (0.939 − 0.342i)29-s + (0.5 − 0.866i)31-s + (−0.766 + 0.642i)33-s + 37-s − 39-s + (0.173 + 0.984i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3944656954 - 1.111027052i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3944656954 - 1.111027052i\) |
\(L(1)\) |
\(\approx\) |
\(0.8378230927 - 0.3300919541i\) |
\(L(1)\) |
\(\approx\) |
\(0.8378230927 - 0.3300919541i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.173 - 0.984i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.173 - 0.984i)T \) |
| 17 | \( 1 + (0.939 + 0.342i)T \) |
| 23 | \( 1 + (0.766 + 0.642i)T \) |
| 29 | \( 1 + (0.939 - 0.342i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.173 + 0.984i)T \) |
| 43 | \( 1 + (-0.766 + 0.642i)T \) |
| 47 | \( 1 + (-0.939 + 0.342i)T \) |
| 53 | \( 1 + (0.766 + 0.642i)T \) |
| 59 | \( 1 + (-0.939 - 0.342i)T \) |
| 61 | \( 1 + (-0.766 - 0.642i)T \) |
| 67 | \( 1 + (0.939 - 0.342i)T \) |
| 71 | \( 1 + (-0.766 + 0.642i)T \) |
| 73 | \( 1 + (-0.173 - 0.984i)T \) |
| 79 | \( 1 + (-0.173 - 0.984i)T \) |
| 83 | \( 1 + (0.5 - 0.866i)T \) |
| 89 | \( 1 + (0.173 - 0.984i)T \) |
| 97 | \( 1 + (0.939 + 0.342i)T \) |
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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
−22.72700207062358478083629187113, −21.53882621711035677440983213984, −21.00588593476273832738472089642, −20.237714686193469152816549538383, −19.501106972900327310060235197160, −18.45368936950662067065116363664, −17.48196663401456050996730641122, −16.63035335158724832929320295807, −16.2184593897995096351911277160, −15.26048186308238019795022042161, −14.39979015172438732293432951750, −13.6955816709394752869038274748, −12.5588860982131195725078677280, −11.733673505264065624432709998111, −10.66748956808420789045490099964, −10.116776385399996770105455477, −9.38654661248258730560427134308, −8.40198990478682474273595383671, −7.195532890569075366301781775213, −6.46356520789533275187944360797, −5.17772575736436430224363641439, −4.49072622391574939545897631723, −3.59220926302952049358816545557, −2.61405024684156517339925017930, −1.00803088619799472224885324362,
0.32841439789373272251460069234, 1.346346843592443599052228350477, 2.77407192799362910521370786652, 3.18150283941278416015868269515, 5.0182322915580546011660435447, 5.90880188649396856637744431146, 6.33649662600876220218052506662, 7.787159334384561713140367057196, 8.14809006773792958569090606485, 9.23866883109826540533004549844, 10.31083251288218530349205084226, 11.33367736391494739328767709188, 12.03435089932030216830649518438, 13.01547659784536569057579606475, 13.31791013786839266866647009407, 14.51767959276159000641420053702, 15.36219528823246302248356274370, 16.28072911891327087712669289725, 17.11243501983613208100942960075, 18.05785648445624106641544057959, 18.67695465287171585493400188334, 19.28476657573392525431195341740, 20.070217055658505561251438880035, 21.25465879033750973037653271710, 21.8536386069584552410212794953