L(s) = 1 | + 2.55·3-s + (1.49 + 1.66i)5-s + (−2.40 − 2.40i)7-s + 3.51·9-s + (2.67 − 2.67i)11-s + 2.40i·13-s + (3.80 + 4.25i)15-s + (−0.0750 − 0.0750i)17-s + (−2.67 + 2.67i)19-s + (−6.13 − 6.13i)21-s + (−2.12 + 2.12i)23-s + (−0.553 + 4.96i)25-s + 1.30·27-s + (−3.95 − 3.95i)29-s + 1.65i·31-s + ⋯ |
L(s) = 1 | + 1.47·3-s + (0.666 + 0.745i)5-s + (−0.908 − 0.908i)7-s + 1.17·9-s + (0.807 − 0.807i)11-s + 0.666i·13-s + (0.982 + 1.09i)15-s + (−0.0182 − 0.0182i)17-s + (−0.613 + 0.613i)19-s + (−1.33 − 1.33i)21-s + (−0.442 + 0.442i)23-s + (−0.110 + 0.993i)25-s + 0.250·27-s + (−0.734 − 0.734i)29-s + 0.297i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.08886 + 0.0823348i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.08886 + 0.0823348i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.49 - 1.66i)T \) |
good | 3 | \( 1 - 2.55T + 3T^{2} \) |
| 7 | \( 1 + (2.40 + 2.40i)T + 7iT^{2} \) |
| 11 | \( 1 + (-2.67 + 2.67i)T - 11iT^{2} \) |
| 13 | \( 1 - 2.40iT - 13T^{2} \) |
| 17 | \( 1 + (0.0750 + 0.0750i)T + 17iT^{2} \) |
| 19 | \( 1 + (2.67 - 2.67i)T - 19iT^{2} \) |
| 23 | \( 1 + (2.12 - 2.12i)T - 23iT^{2} \) |
| 29 | \( 1 + (3.95 + 3.95i)T + 29iT^{2} \) |
| 31 | \( 1 - 1.65iT - 31T^{2} \) |
| 37 | \( 1 + 2.53iT - 37T^{2} \) |
| 41 | \( 1 - 1.70iT - 41T^{2} \) |
| 43 | \( 1 + 3.84iT - 43T^{2} \) |
| 47 | \( 1 + (2.15 - 2.15i)T - 47iT^{2} \) |
| 53 | \( 1 + 1.29T + 53T^{2} \) |
| 59 | \( 1 + (5.29 + 5.29i)T + 59iT^{2} \) |
| 61 | \( 1 + (-10.2 + 10.2i)T - 61iT^{2} \) |
| 67 | \( 1 - 10.6iT - 67T^{2} \) |
| 71 | \( 1 + 2.27T + 71T^{2} \) |
| 73 | \( 1 + (9.99 + 9.99i)T + 73iT^{2} \) |
| 79 | \( 1 - 8.70T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 + (-5.00 - 5.00i)T + 97iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.53459733277683524789433076321, −10.43571369802597599988390648011, −9.634845578926620154479143434212, −9.015945433515817079183199543803, −7.86058161406581482399959104383, −6.87096333753784739827198184415, −6.06919615972420192376177274116, −3.92449237056326728513305060767, −3.32844435439775515044319410858, −1.96319896342850022954278846408,
1.95682592277266615450347428663, 2.98326779501907210682332033398, 4.32294621332661723556169271232, 5.72772877516656353812590080226, 6.83867520815661248128660687444, 8.149903749359742582756358359353, 9.007349288966150565036381189372, 9.386204735066365053847369056728, 10.25758878896079888997918496619, 11.95271674471770141718719149382