| L(s) = 1 | + 2i·2-s + 3i·3-s − 4·4-s − 6·6-s − 14i·7-s − 8i·8-s − 9·9-s + 11·11-s − 12i·12-s + 80i·13-s + 28·14-s + 16·16-s − 30i·17-s − 18i·18-s − 56·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.755i·7-s − 0.353i·8-s − 0.333·9-s + 0.301·11-s − 0.288i·12-s + 1.70i·13-s + 0.534·14-s + 0.250·16-s − 0.428i·17-s − 0.235i·18-s − 0.676·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.728882140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.728882140\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2iT \) |
| 3 | \( 1 - 3iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 7 | \( 1 + 14iT - 343T^{2} \) |
| 13 | \( 1 - 80iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 30iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 56T + 6.85e3T^{2} \) |
| 23 | \( 1 + 126iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 222T + 2.43e4T^{2} \) |
| 31 | \( 1 + 16T + 2.97e4T^{2} \) |
| 37 | \( 1 - 106iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 114T + 6.89e4T^{2} \) |
| 43 | \( 1 + 52iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 246iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 264iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 264T + 2.05e5T^{2} \) |
| 61 | \( 1 - 92T + 2.26e5T^{2} \) |
| 67 | \( 1 - 796iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 426T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.17e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 842T + 4.93e5T^{2} \) |
| 83 | \( 1 - 852iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.06e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.28e3iT - 9.12e5T^{2} \) |
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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
−9.048847157922289475938266027911, −8.643771129578116357788453522612, −7.60254680931525930687531715511, −6.67289744221264786608868768271, −6.35375210643377462112185657272, −4.95771324866655246549777336275, −4.40668024140145834066928344927, −3.70348970865567097435138853100, −2.29824213149477192691701130071, −0.818296200993271906033557257458,
0.50765480275322584495882246250, 1.56901505720109754669667902620, 2.63249424252351279769863459030, 3.33193754892157837471457175350, 4.53673405742854260207416583782, 5.61185215727687745192037078589, 6.08261916094949223670679463313, 7.30887240786027889982670477447, 8.143070786874588471232525199361, 8.689223038619894359470950896460
