L(s) = 1 | + (0.441 + 0.897i)2-s + (0.700 + 0.714i)3-s + (−0.610 + 0.792i)4-s + (0.921 + 0.387i)5-s + (−0.331 + 0.943i)6-s + (0.641 − 0.767i)7-s + (−0.980 − 0.197i)8-s + (−0.0198 + 0.999i)9-s + (0.0596 + 0.998i)10-s + (0.971 + 0.236i)11-s + (−0.992 + 0.119i)12-s + (−0.827 + 0.561i)13-s + (0.971 + 0.236i)14-s + (0.368 + 0.929i)15-s + (−0.255 − 0.966i)16-s + (−0.780 − 0.625i)17-s + ⋯ |
L(s) = 1 | + (0.441 + 0.897i)2-s + (0.700 + 0.714i)3-s + (−0.610 + 0.792i)4-s + (0.921 + 0.387i)5-s + (−0.331 + 0.943i)6-s + (0.641 − 0.767i)7-s + (−0.980 − 0.197i)8-s + (−0.0198 + 0.999i)9-s + (0.0596 + 0.998i)10-s + (0.971 + 0.236i)11-s + (−0.992 + 0.119i)12-s + (−0.827 + 0.561i)13-s + (0.971 + 0.236i)14-s + (0.368 + 0.929i)15-s + (−0.255 − 0.966i)16-s + (−0.780 − 0.625i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.018207476 + 1.966997679i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.018207476 + 1.966997679i\) |
\(L(1)\) |
\(\approx\) |
\(1.270988727 + 1.220643101i\) |
\(L(1)\) |
\(\approx\) |
\(1.270988727 + 1.220643101i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 317 | \( 1 \) |
good | 2 | \( 1 + (0.441 + 0.897i)T \) |
| 3 | \( 1 + (0.700 + 0.714i)T \) |
| 5 | \( 1 + (0.921 + 0.387i)T \) |
| 7 | \( 1 + (0.641 - 0.767i)T \) |
| 11 | \( 1 + (0.971 + 0.236i)T \) |
| 13 | \( 1 + (-0.827 + 0.561i)T \) |
| 17 | \( 1 + (-0.780 - 0.625i)T \) |
| 19 | \( 1 + (-0.331 - 0.943i)T \) |
| 23 | \( 1 + (0.987 + 0.158i)T \) |
| 29 | \( 1 + (0.641 + 0.767i)T \) |
| 31 | \( 1 + (-0.727 - 0.685i)T \) |
| 37 | \( 1 + (-0.961 + 0.274i)T \) |
| 41 | \( 1 + (-0.999 - 0.0397i)T \) |
| 43 | \( 1 + (0.578 - 0.815i)T \) |
| 47 | \( 1 + (-0.727 - 0.685i)T \) |
| 53 | \( 1 + (-0.331 - 0.943i)T \) |
| 59 | \( 1 + (0.578 + 0.815i)T \) |
| 61 | \( 1 + (-0.869 + 0.494i)T \) |
| 67 | \( 1 + (-0.671 + 0.741i)T \) |
| 71 | \( 1 + (0.368 - 0.929i)T \) |
| 73 | \( 1 + (0.138 - 0.990i)T \) |
| 79 | \( 1 + (-0.610 - 0.792i)T \) |
| 83 | \( 1 + (0.511 - 0.859i)T \) |
| 89 | \( 1 + (0.441 + 0.897i)T \) |
| 97 | \( 1 + (-0.961 + 0.274i)T \) |
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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
−24.655468510459787359435614495998, −24.296071760261380648851135502905, −22.96033958518911787114393221064, −21.89577927855074148065059035583, −21.26484805813085735835418616450, −20.43390995282497731565713071828, −19.565827437293631329870071650412, −18.810378984445087648951315298854, −17.79319752122595902976320935014, −17.22737791478754089533124355364, −15.23358012408517948982953622096, −14.4839871803812915486657179269, −13.87585160985322578396589643869, −12.66338696893759240022864614835, −12.38004372437642370974620873772, −11.131197501864853763101871510632, −9.85904195344683810143811136769, −8.97967400449419318284511814977, −8.3268118568293144543353361068, −6.565000263045633662522397770877, −5.6622817647556207234308923445, −4.51676790559772646857775981945, −3.09330148772023531933013959418, −2.06116449729604946427978234553, −1.360164906385683066172168936174,
2.05125996038942982344011983522, 3.32873776398852142039254986001, 4.540755305209721008203975369680, 5.099280870657799900880644432999, 6.77717324364765802115842261415, 7.25089547662905223910304715841, 8.80164454189965139375807304057, 9.30170695388561721533762842298, 10.46042018968613646022817646094, 11.62882544343799761703510732528, 13.25661747175627171079097538873, 13.83786806041378539445832375320, 14.6270621419933008464135570878, 15.12889660759611023264814477550, 16.511485362182819121189631858153, 17.156820055772682020906863940861, 17.88130469472368197765752111860, 19.287895125535903207436478509769, 20.37940545731220163284998260409, 21.24779469060557416150177828795, 22.02341651122824898526680367502, 22.56225620253679741435667006066, 23.95548149845898090608177264798, 24.698280571515008765433263909020, 25.50413887377836198815629305181