L(s) = 1 | + (0.959 − 0.281i)3-s + (0.989 − 0.142i)7-s + (0.841 − 0.540i)9-s + (0.909 + 0.415i)11-s + (−0.142 + 0.989i)13-s + (0.755 − 0.654i)17-s + (−0.755 − 0.654i)19-s + (0.909 − 0.415i)21-s + (0.654 − 0.755i)27-s + (−0.755 + 0.654i)29-s + (0.959 + 0.281i)31-s + (0.989 + 0.142i)33-s + (0.841 − 0.540i)37-s + (0.142 + 0.989i)39-s + (−0.841 − 0.540i)41-s + ⋯ |
L(s) = 1 | + (0.959 − 0.281i)3-s + (0.989 − 0.142i)7-s + (0.841 − 0.540i)9-s + (0.909 + 0.415i)11-s + (−0.142 + 0.989i)13-s + (0.755 − 0.654i)17-s + (−0.755 − 0.654i)19-s + (0.909 − 0.415i)21-s + (0.654 − 0.755i)27-s + (−0.755 + 0.654i)29-s + (0.959 + 0.281i)31-s + (0.989 + 0.142i)33-s + (0.841 − 0.540i)37-s + (0.142 + 0.989i)39-s + (−0.841 − 0.540i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.921768054 - 0.4392502959i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.921768054 - 0.4392502959i\) |
\(L(1)\) |
\(\approx\) |
\(1.760334070 - 0.1656522490i\) |
\(L(1)\) |
\(\approx\) |
\(1.760334070 - 0.1656522490i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (0.959 - 0.281i)T \) |
| 7 | \( 1 + (0.989 - 0.142i)T \) |
| 11 | \( 1 + (0.909 + 0.415i)T \) |
| 13 | \( 1 + (-0.142 + 0.989i)T \) |
| 17 | \( 1 + (0.755 - 0.654i)T \) |
| 19 | \( 1 + (-0.755 - 0.654i)T \) |
| 29 | \( 1 + (-0.755 + 0.654i)T \) |
| 31 | \( 1 + (0.959 + 0.281i)T \) |
| 37 | \( 1 + (0.841 - 0.540i)T \) |
| 41 | \( 1 + (-0.841 - 0.540i)T \) |
| 43 | \( 1 + (-0.959 + 0.281i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (0.142 + 0.989i)T \) |
| 59 | \( 1 + (0.989 + 0.142i)T \) |
| 61 | \( 1 + (0.281 - 0.959i)T \) |
| 67 | \( 1 + (0.415 + 0.909i)T \) |
| 71 | \( 1 + (0.415 + 0.909i)T \) |
| 73 | \( 1 + (-0.755 - 0.654i)T \) |
| 79 | \( 1 + (-0.142 + 0.989i)T \) |
| 83 | \( 1 + (-0.841 + 0.540i)T \) |
| 89 | \( 1 + (-0.959 + 0.281i)T \) |
| 97 | \( 1 + (-0.540 + 0.841i)T \) |
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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
−20.31199810808272479737772666843, −19.4038037377830389566434693436, −18.91682412498851014971113882035, −18.06888484773974463112090004523, −17.11864752506854034909958016018, −16.63164344568437285531872797661, −15.475307815456905067314062675982, −14.82183836323441711074415635421, −14.54927070074192524484039157860, −13.606610158741061437587114946103, −12.89757219593515162100092945201, −11.96912737609568892499119816421, −11.19017333957484908647877608155, −10.24015053019958016472406152000, −9.73116155606994746888104768212, −8.55749332785168898428134357590, −8.236216109865250365301646293937, −7.563169564274169351126566794289, −6.37909100563616498481971210555, −5.505828584381991865452568063030, −4.51878916201201195732256829073, −3.80565531127162240066287429117, −2.97784902217042800250674077108, −1.94453097566373084639320742543, −1.17865928206565280562312290604,
1.13062012620892667747048130715, 1.85446475997169841387796796051, 2.67418041770457206561305163370, 3.855706945273047865848982521882, 4.40131292361797283860814576132, 5.34241114346057018290195998019, 6.7845583547036100834131256673, 7.0431153190103130211971243088, 8.07145235358184146450325396618, 8.74004738952044985787153296066, 9.415882676232586683289895891259, 10.181502271871929852272767258524, 11.336019370108552543525725944293, 11.88311942725402426480065586331, 12.7210649273044846424213126125, 13.68225288216672884887249798596, 14.24512830288932894921096103370, 14.76478808814936396888552248255, 15.42776580797578142135639460444, 16.555075271891473166511607977015, 17.17657810408263238480827621673, 18.05755153682957523233937187224, 18.70472338305149579832467806879, 19.42119746542955802391707886207, 20.13580778140325189924588009016