| L(s) = 1 | + i·3-s − i·5-s + 2.40·7-s − 9-s − 2.34·11-s − 5.36·13-s + 15-s + 0.425i·17-s + 3.47·19-s + 2.40i·21-s + (−0.874 + 4.71i)23-s − 25-s − i·27-s + 4.06·29-s − 10.3i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.447i·5-s + 0.908·7-s − 0.333·9-s − 0.706·11-s − 1.48·13-s + 0.258·15-s + 0.103i·17-s + 0.796·19-s + 0.524i·21-s + (−0.182 + 0.983i)23-s − 0.200·25-s − 0.192i·27-s + 0.754·29-s − 1.86i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.760 - 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9112290687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9112290687\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (0.874 - 4.71i)T \) |
| good | 7 | \( 1 - 2.40T + 7T^{2} \) |
| 11 | \( 1 + 2.34T + 11T^{2} \) |
| 13 | \( 1 + 5.36T + 13T^{2} \) |
| 17 | \( 1 - 0.425iT - 17T^{2} \) |
| 19 | \( 1 - 3.47T + 19T^{2} \) |
| 29 | \( 1 - 4.06T + 29T^{2} \) |
| 31 | \( 1 + 10.3iT - 31T^{2} \) |
| 37 | \( 1 - 5.77iT - 37T^{2} \) |
| 41 | \( 1 - 1.57T + 41T^{2} \) |
| 43 | \( 1 - 5.79T + 43T^{2} \) |
| 47 | \( 1 - 9.78iT - 47T^{2} \) |
| 53 | \( 1 - 1.33iT - 53T^{2} \) |
| 59 | \( 1 - 9.09iT - 59T^{2} \) |
| 61 | \( 1 - 7.73iT - 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + 11.0iT - 71T^{2} \) |
| 73 | \( 1 + 6.28T + 73T^{2} \) |
| 79 | \( 1 + 8.31T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 - 3.47iT - 89T^{2} \) |
| 97 | \( 1 + 3.91iT - 97T^{2} \) |
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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
−8.372507997351841004800705209995, −7.59699919553218245712196727852, −7.44027914546419644197742525949, −5.97637277497174960277319065705, −5.48317936603364036876465807280, −4.61717466966251292824250806764, −4.39513865312710688672921194901, −3.07082325214588287934072741564, −2.35879233061851824437654158133, −1.19939900122714406943418479449,
0.23722400293374122205364205555, 1.57188300740124419699239447291, 2.48370566465154501702887615650, 3.04985820522801021549645129825, 4.34242790431866248483142467741, 5.08215074821933974589083185997, 5.57253464981491922990865141278, 6.72079732311890910138731358163, 7.16860880745461560833056203881, 7.81495366151751287683192303968
