| L(s) = 1 | − 3-s − 5-s − 0.549i·7-s + 9-s + 1.98i·11-s − 2.38i·13-s + 15-s − 4.49·17-s + (4.14 − 1.35i)19-s + 0.549i·21-s + 2.88i·23-s + 25-s − 27-s + 3.54i·29-s − 0.0281·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.207i·7-s + 0.333·9-s + 0.598i·11-s − 0.662i·13-s + 0.258·15-s − 1.09·17-s + (0.950 − 0.309i)19-s + 0.119i·21-s + 0.602i·23-s + 0.200·25-s − 0.192·27-s + 0.658i·29-s − 0.00505·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.207 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7671886164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7671886164\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + (-4.14 + 1.35i)T \) |
| good | 7 | \( 1 + 0.549iT - 7T^{2} \) |
| 11 | \( 1 - 1.98iT - 11T^{2} \) |
| 13 | \( 1 + 2.38iT - 13T^{2} \) |
| 17 | \( 1 + 4.49T + 17T^{2} \) |
| 23 | \( 1 - 2.88iT - 23T^{2} \) |
| 29 | \( 1 - 3.54iT - 29T^{2} \) |
| 31 | \( 1 + 0.0281T + 31T^{2} \) |
| 37 | \( 1 + 3.37iT - 37T^{2} \) |
| 41 | \( 1 - 4.40iT - 41T^{2} \) |
| 43 | \( 1 + 6.83iT - 43T^{2} \) |
| 47 | \( 1 + 10.4iT - 47T^{2} \) |
| 53 | \( 1 - 12.4iT - 53T^{2} \) |
| 59 | \( 1 + 3.79T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 4.92T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 + 7.76T + 73T^{2} \) |
| 79 | \( 1 + 2.04T + 79T^{2} \) |
| 83 | \( 1 + 7.95iT - 83T^{2} \) |
| 89 | \( 1 + 0.0323iT - 89T^{2} \) |
| 97 | \( 1 - 0.596iT - 97T^{2} \) |
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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
−7.983496087550026439933975846280, −7.25326763735707761727761847810, −6.87607118015562251445640211202, −5.83228673651414884371739759917, −5.17558239248879165232573119710, −4.43652758754577646168831327122, −3.65038683508816604440804887748, −2.66691535578999955488356390593, −1.48077940564034242118764581388, −0.28156280992094712674984476123,
0.966201378315033151491380872268, 2.19117955956738596588901085078, 3.21244040115098372129464509624, 4.16281811453144413978376403844, 4.76601813086914914895679510349, 5.66127178125726646377328015852, 6.38135408256221940281179600776, 6.95579585717995101696520106217, 7.84238790630200646399414468264, 8.460792067778739445761355018531
