L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s − 11-s − 13-s − 14-s + 16-s − 17-s + 19-s + 20-s + 22-s + 23-s + 25-s + 26-s + 28-s + 29-s − 31-s − 32-s + 34-s + 35-s + 37-s − 38-s − 40-s + 41-s + ⋯ |
L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s − 11-s − 13-s − 14-s + 16-s − 17-s + 19-s + 20-s + 22-s + 23-s + 25-s + 26-s + 28-s + 29-s − 31-s − 32-s + 34-s + 35-s + 37-s − 38-s − 40-s + 41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 411 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 411 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.626467650\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.626467650\) |
\(L(1)\) |
\(\approx\) |
\(0.9297800258\) |
\(L(1)\) |
\(\approx\) |
\(0.9297800258\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 137 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 - T \) |
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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
−24.499101606547688483127997506376, −23.49594462544514785994409965374, −21.97187053423888882291495938026, −21.349974622299469165585684232648, −20.53237553474074335291959435686, −19.80608556280237026677086296678, −18.49349075719317787211220147705, −17.97716115404585280618116997501, −17.33366546494640101920810959704, −16.45934259289122820484133581544, −15.340412916882108286169453842723, −14.55363590507511571587946572115, −13.46079757362801050840175668096, −12.375906420822041777329650365414, −11.226312386498288902401453224287, −10.53593875442848032545929262263, −9.60587821067805894594406266452, −8.78489019475498731965836300710, −7.72443506001733004113471737618, −6.93584810436780098614920595491, −5.63406437257690271759422306837, −4.840081861849494663833741353535, −2.77328403999234838611661725137, −2.078118301775149456746025867704, −0.83818836870164467450936180208,
0.83818836870164467450936180208, 2.078118301775149456746025867704, 2.77328403999234838611661725137, 4.840081861849494663833741353535, 5.63406437257690271759422306837, 6.93584810436780098614920595491, 7.72443506001733004113471737618, 8.78489019475498731965836300710, 9.60587821067805894594406266452, 10.53593875442848032545929262263, 11.226312386498288902401453224287, 12.375906420822041777329650365414, 13.46079757362801050840175668096, 14.55363590507511571587946572115, 15.340412916882108286169453842723, 16.45934259289122820484133581544, 17.33366546494640101920810959704, 17.97716115404585280618116997501, 18.49349075719317787211220147705, 19.80608556280237026677086296678, 20.53237553474074335291959435686, 21.349974622299469165585684232648, 21.97187053423888882291495938026, 23.49594462544514785994409965374, 24.499101606547688483127997506376