L(s) = 1 | + 0.237i·3-s + 5-s + (2.63 − 0.174i)7-s + 2.94·9-s − 4.69·11-s − 2.77·13-s + 0.237i·15-s − 3.55i·17-s − 6.72i·19-s + (0.0415 + 0.627i)21-s − 8.45i·23-s + 25-s + 1.41i·27-s − 5.80i·29-s − 6.92·31-s + ⋯ |
L(s) = 1 | + 0.137i·3-s + 0.447·5-s + (0.997 − 0.0660i)7-s + 0.981·9-s − 1.41·11-s − 0.770·13-s + 0.0613i·15-s − 0.862i·17-s − 1.54i·19-s + (0.00906 + 0.136i)21-s − 1.76i·23-s + 0.200·25-s + 0.271i·27-s − 1.07i·29-s − 1.24·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.797574491\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.797574491\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-2.63 + 0.174i)T \) |
good | 3 | \( 1 - 0.237iT - 3T^{2} \) |
| 11 | \( 1 + 4.69T + 11T^{2} \) |
| 13 | \( 1 + 2.77T + 13T^{2} \) |
| 17 | \( 1 + 3.55iT - 17T^{2} \) |
| 19 | \( 1 + 6.72iT - 19T^{2} \) |
| 23 | \( 1 + 8.45iT - 23T^{2} \) |
| 29 | \( 1 + 5.80iT - 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 + 0.171iT - 37T^{2} \) |
| 41 | \( 1 - 5.71iT - 41T^{2} \) |
| 43 | \( 1 - 3.26T + 43T^{2} \) |
| 47 | \( 1 - 2.83T + 47T^{2} \) |
| 53 | \( 1 + 7.09iT - 53T^{2} \) |
| 59 | \( 1 - 4.79iT - 59T^{2} \) |
| 61 | \( 1 - 9.63T + 61T^{2} \) |
| 67 | \( 1 + 9.99T + 67T^{2} \) |
| 71 | \( 1 + 7.49iT - 71T^{2} \) |
| 73 | \( 1 - 8.35iT - 73T^{2} \) |
| 79 | \( 1 + 5.05iT - 79T^{2} \) |
| 83 | \( 1 - 9.53iT - 83T^{2} \) |
| 89 | \( 1 + 5.75iT - 89T^{2} \) |
| 97 | \( 1 - 12.7iT - 97T^{2} \) |
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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
−8.935502400123607358534644858979, −8.014762330467503157528424880614, −7.36836638291588876263557218163, −6.73353379212525243888654329679, −5.46327137394406108979671922401, −4.85064140335583946184055262655, −4.33943697454715568690470789114, −2.70571668265045876714948691652, −2.16404833799860230306508875078, −0.60424111938105952612693129595,
1.50629821028149035085061442204, 2.08587745411794682112922973658, 3.45613487030780816808566488735, 4.43229740779130305063050035854, 5.42473201547141782691639224392, 5.71483997200939121276538209297, 7.20074354665106214276749474179, 7.55579415691769519391715650944, 8.262552919233885405308157040145, 9.221471259295088494882736167453