L(s) = 1 | + (−0.719 − 0.694i)2-s + (0.848 + 0.529i)3-s + (0.0348 + 0.999i)4-s + (−0.241 − 0.970i)6-s + (−0.669 − 0.743i)7-s + (0.669 − 0.743i)8-s + (0.438 + 0.898i)9-s + (−0.5 + 0.866i)12-s + (−0.997 − 0.0697i)13-s + (−0.0348 + 0.999i)14-s + (−0.997 + 0.0697i)16-s + (−0.438 + 0.898i)17-s + (0.309 − 0.951i)18-s + (−0.173 − 0.984i)21-s + (0.939 − 0.342i)23-s + (0.961 − 0.275i)24-s + ⋯ |
L(s) = 1 | + (−0.719 − 0.694i)2-s + (0.848 + 0.529i)3-s + (0.0348 + 0.999i)4-s + (−0.241 − 0.970i)6-s + (−0.669 − 0.743i)7-s + (0.669 − 0.743i)8-s + (0.438 + 0.898i)9-s + (−0.5 + 0.866i)12-s + (−0.997 − 0.0697i)13-s + (−0.0348 + 0.999i)14-s + (−0.997 + 0.0697i)16-s + (−0.438 + 0.898i)17-s + (0.309 − 0.951i)18-s + (−0.173 − 0.984i)21-s + (0.939 − 0.342i)23-s + (0.961 − 0.275i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.05601798989 - 0.3274509420i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05601798989 - 0.3274509420i\) |
\(L(1)\) |
\(\approx\) |
\(0.7799547322 - 0.09797959242i\) |
\(L(1)\) |
\(\approx\) |
\(0.7799547322 - 0.09797959242i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.719 - 0.694i)T \) |
| 3 | \( 1 + (0.848 + 0.529i)T \) |
| 7 | \( 1 + (-0.669 - 0.743i)T \) |
| 13 | \( 1 + (-0.997 - 0.0697i)T \) |
| 17 | \( 1 + (-0.438 + 0.898i)T \) |
| 23 | \( 1 + (0.939 - 0.342i)T \) |
| 29 | \( 1 + (0.882 + 0.469i)T \) |
| 31 | \( 1 + (0.104 + 0.994i)T \) |
| 37 | \( 1 + (0.309 - 0.951i)T \) |
| 41 | \( 1 + (-0.848 - 0.529i)T \) |
| 43 | \( 1 + (0.939 + 0.342i)T \) |
| 47 | \( 1 + (-0.990 + 0.139i)T \) |
| 53 | \( 1 + (0.559 - 0.829i)T \) |
| 59 | \( 1 + (-0.990 - 0.139i)T \) |
| 61 | \( 1 + (0.961 + 0.275i)T \) |
| 67 | \( 1 + (0.173 - 0.984i)T \) |
| 71 | \( 1 + (-0.559 - 0.829i)T \) |
| 73 | \( 1 + (0.374 + 0.927i)T \) |
| 79 | \( 1 + (0.241 - 0.970i)T \) |
| 83 | \( 1 + (-0.913 + 0.406i)T \) |
| 89 | \( 1 + (-0.766 - 0.642i)T \) |
| 97 | \( 1 + (-0.719 - 0.694i)T \) |
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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
−21.697545687329870989528534937861, −20.589302877673146272878603074147, −19.874058430882964570598192515873, −19.149875880422541388390499775446, −18.71988794151638188074667429761, −17.893229713068195123892106968560, −17.100135383114759623365977185253, −16.14698594085350174619701599281, −15.277943269131269778796692634119, −14.93534505450071374352505105641, −13.87732094341542585630952535635, −13.24300783101187196158276406097, −12.211999601342568244486818483165, −11.38252246686778947634652364569, −9.92704357911320430478809747491, −9.538406842446430078234209787645, −8.76104154735707551632670315908, −7.98027371292710101852168336269, −7.07050723828981073490337543219, −6.55044816453685014855815997989, −5.48756480910565159536439798715, −4.45299251652268186706315130701, −2.88921739194914557614933766248, −2.338928502114412740792657948609, −1.056272811798418726329719430060,
0.083366780221276245044319897593, 1.40936650008209655905561899140, 2.522957009801303194656858699529, 3.24828391170130159009051392872, 4.0902266877361374111331564713, 4.92215797734789892148419780253, 6.69040703785359531534918576873, 7.35592303781597139612229277290, 8.293231351492893713092677966393, 9.02281383139680799398560460547, 9.81471036876191079503245415937, 10.43432951494446365117418863992, 11.0350627602054031144129480575, 12.42256255988198782549982222604, 12.91894652812996105103771911675, 13.82199962799797202709298643658, 14.67405406288309615468967462914, 15.67462909432597910887386935514, 16.42229915355910903198771005942, 17.09510554171146128880267353146, 17.9077773643412214698728518171, 19.097228441967129860887773439641, 19.57335712273685273610888645263, 19.96701007354767559769019129458, 20.90067302277786123698419288683