| L(s) = 1 | + 0.936·3-s − i·5-s + (2.60 − 0.468i)7-s − 2.12·9-s + 2.39i·11-s − 2i·13-s − 0.936i·15-s − 7.12i·17-s + 2.39·19-s + (2.43 − 0.438i)21-s − 5.73i·23-s − 25-s − 4.79·27-s + 2·29-s − 6.67·31-s + ⋯ |
| L(s) = 1 | + 0.540·3-s − 0.447i·5-s + (0.984 − 0.176i)7-s − 0.707·9-s + 0.723i·11-s − 0.554i·13-s − 0.241i·15-s − 1.72i·17-s + 0.550·19-s + (0.532 − 0.0956i)21-s − 1.19i·23-s − 0.200·25-s − 0.923·27-s + 0.371·29-s − 1.19·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.176 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.176 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.014987560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.014987560\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (-2.60 + 0.468i)T \) |
| good | 3 | \( 1 - 0.936T + 3T^{2} \) |
| 11 | \( 1 - 2.39iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 7.12iT - 17T^{2} \) |
| 19 | \( 1 - 2.39T + 19T^{2} \) |
| 23 | \( 1 + 5.73iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 6.67T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 7.12iT - 41T^{2} \) |
| 43 | \( 1 + 7.60iT - 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 + 14.2iT - 67T^{2} \) |
| 71 | \( 1 + 6.14iT - 71T^{2} \) |
| 73 | \( 1 + 9.36iT - 73T^{2} \) |
| 79 | \( 1 - 4.27iT - 79T^{2} \) |
| 83 | \( 1 - 0.936T + 83T^{2} \) |
| 89 | \( 1 - 12iT - 89T^{2} \) |
| 97 | \( 1 - 7.12iT - 97T^{2} \) |
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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
−8.947494271378196166717441324645, −8.010487836812149222238130355418, −7.59083287521610577429648028513, −6.67837137559153672129491955185, −5.34598066798388730697975242772, −5.03478066301450906124919744646, −4.00223197526663494967005825920, −2.87323268773115603182658723468, −2.05510125885666237789316120808, −0.65026121636732610196920183241,
1.46573518066035793925626668171, 2.40938064840738184691841860154, 3.47272421971942313459534166513, 4.12949656069635031972357136487, 5.54368812354757254127603521792, 5.80771057889085731202292215801, 7.05602600654775154552960953941, 7.77402662119586630173749776450, 8.552500988632049997958039005873, 8.892376486306455043004763020961
