L(s) = 1 | − 1.96·3-s + (−1.42 + 1.72i)5-s + (1.60 − 1.60i)7-s + 0.851·9-s + (−0.754 − 0.754i)11-s − 5.94i·13-s + (2.79 − 3.38i)15-s + (1.95 − 1.95i)17-s + (−0.780 − 0.780i)19-s + (−3.14 + 3.14i)21-s + (−4.93 − 4.93i)23-s + (−0.956 − 4.90i)25-s + 4.21·27-s + (−1.44 + 1.44i)29-s − 3.60i·31-s + ⋯ |
L(s) = 1 | − 1.13·3-s + (−0.635 + 0.771i)5-s + (0.605 − 0.605i)7-s + 0.283·9-s + (−0.227 − 0.227i)11-s − 1.64i·13-s + (0.720 − 0.874i)15-s + (0.474 − 0.474i)17-s + (−0.179 − 0.179i)19-s + (−0.686 + 0.686i)21-s + (−1.02 − 1.02i)23-s + (−0.191 − 0.981i)25-s + 0.811·27-s + (−0.268 + 0.268i)29-s − 0.648i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.105 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.370303 - 0.411597i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.370303 - 0.411597i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.42 - 1.72i)T \) |
good | 3 | \( 1 + 1.96T + 3T^{2} \) |
| 7 | \( 1 + (-1.60 + 1.60i)T - 7iT^{2} \) |
| 11 | \( 1 + (0.754 + 0.754i)T + 11iT^{2} \) |
| 13 | \( 1 + 5.94iT - 13T^{2} \) |
| 17 | \( 1 + (-1.95 + 1.95i)T - 17iT^{2} \) |
| 19 | \( 1 + (0.780 + 0.780i)T + 19iT^{2} \) |
| 23 | \( 1 + (4.93 + 4.93i)T + 23iT^{2} \) |
| 29 | \( 1 + (1.44 - 1.44i)T - 29iT^{2} \) |
| 31 | \( 1 + 3.60iT - 31T^{2} \) |
| 37 | \( 1 + 10.2iT - 37T^{2} \) |
| 41 | \( 1 - 6.93iT - 41T^{2} \) |
| 43 | \( 1 - 9.91iT - 43T^{2} \) |
| 47 | \( 1 + (0.104 + 0.104i)T + 47iT^{2} \) |
| 53 | \( 1 + 4.03T + 53T^{2} \) |
| 59 | \( 1 + (-3.46 + 3.46i)T - 59iT^{2} \) |
| 61 | \( 1 + (-0.680 - 0.680i)T + 61iT^{2} \) |
| 67 | \( 1 + 9.04iT - 67T^{2} \) |
| 71 | \( 1 - 3.64T + 71T^{2} \) |
| 73 | \( 1 + (2.94 - 2.94i)T - 73iT^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 4.23T + 83T^{2} \) |
| 89 | \( 1 - 0.0426T + 89T^{2} \) |
| 97 | \( 1 + (1.91 - 1.91i)T - 97iT^{2} \) |
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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
−11.10911444596315035794152007711, −10.83421511985559487318727931423, −9.945251890226173407676726575126, −8.164018404537016423693842628215, −7.60689032558301287893876369866, −6.39442261507817503371376756824, −5.49289750678743328764150694271, −4.35035839528706209602024411318, −2.93971134100686182953873611210, −0.46434136903488373139828597067,
1.69408532289789952078395533352, 3.97722880222003028260017831831, 5.00443646359248676940071172940, 5.77575997328514084510818646789, 6.97657273947851703312028989917, 8.174710870156946724906348462872, 8.970937094551269898120429467572, 10.15610276823732529240642064897, 11.32950457009820379925438871065, 11.89920671154250037037499299298