L(s) = 1 | + 3-s − 1.73i·5-s − 2·9-s + 1.73i·11-s − 1.73i·15-s + 5.19i·17-s + 7·19-s − 8.66i·23-s + 2.00·25-s − 5·27-s + 6·29-s − 5·31-s + 1.73i·33-s + 5·37-s − 6.92i·41-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.774i·5-s − 0.666·9-s + 0.522i·11-s − 0.447i·15-s + 1.26i·17-s + 1.60·19-s − 1.80i·23-s + 0.400·25-s − 0.962·27-s + 1.11·29-s − 0.898·31-s + 0.301i·33-s + 0.821·37-s − 1.08i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.163325257\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.163325257\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 11 | \( 1 - 1.73iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 5.19iT - 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 + 8.66iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + 6.92iT - 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 - 9T + 59T^{2} \) |
| 61 | \( 1 + 8.66iT - 61T^{2} \) |
| 67 | \( 1 + 5.19iT - 67T^{2} \) |
| 71 | \( 1 - 3.46iT - 71T^{2} \) |
| 73 | \( 1 + 1.73iT - 73T^{2} \) |
| 79 | \( 1 - 5.19iT - 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + 12.1iT - 89T^{2} \) |
| 97 | \( 1 + 6.92iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.546385384589284479982056641734, −8.115215949515484345754467513981, −7.21812120254270693857184376539, −6.31838948828597061911485155841, −5.46943728520672353586404254574, −4.73809071068453023443969981513, −3.86074215623276887522166577220, −2.92601914944708156330236485394, −2.00006622248901601330974909776, −0.76042203092610399552418267737,
1.01885239576130876012187386892, 2.52918105896530320185715706917, 3.08562214044620319554729449932, 3.72643064003023099701042142515, 5.13061066757583265920143260829, 5.62329586144531152336108458573, 6.65115691311927965064266475354, 7.40962395488731866249546064208, 7.88852384153960572650097579682, 8.871112887932433036668363085922