| L(s) = 1 | + 2i·2-s + 3i·3-s − 4·4-s − 6·6-s − 32i·7-s − 8i·8-s − 9·9-s − 60·11-s − 12i·12-s − 34i·13-s + 64·14-s + 16·16-s − 42i·17-s − 18i·18-s + 76·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 1.72i·7-s − 0.353i·8-s − 0.333·9-s − 1.64·11-s − 0.288i·12-s − 0.725i·13-s + 1.22·14-s + 0.250·16-s − 0.599i·17-s − 0.235i·18-s + 0.917·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.700717 - 0.433067i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.700717 - 0.433067i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 32iT - 343T^{2} \) |
| 11 | \( 1 + 60T + 1.33e3T^{2} \) |
| 13 | \( 1 + 34iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 42iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 76T + 6.85e3T^{2} \) |
| 23 | \( 1 - 1.21e4T^{2} \) |
| 29 | \( 1 + 6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 232T + 2.97e4T^{2} \) |
| 37 | \( 1 + 134iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 234T + 6.89e4T^{2} \) |
| 43 | \( 1 + 412iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 360iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 222iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 660T + 2.05e5T^{2} \) |
| 61 | \( 1 + 490T + 2.26e5T^{2} \) |
| 67 | \( 1 + 812iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 120T + 3.57e5T^{2} \) |
| 73 | \( 1 - 746iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 152T + 4.93e5T^{2} \) |
| 83 | \( 1 + 804iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 678T + 7.04e5T^{2} \) |
| 97 | \( 1 + 194iT - 9.12e5T^{2} \) |
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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
−12.70074530777439346284895072078, −10.91310889903058461504184134554, −10.39420403241401402880704709064, −9.341902598557670757364025634196, −7.77435807833639952018293444663, −7.36434527055723895621211508998, −5.63854425080443732257268933434, −4.63354267210361173861952079542, −3.29529185605881722530371142892, −0.38565541424593073791654779510,
1.94805511515521008373367675398, 2.98999304149075140160991469129, 5.06377722316938025968189036615, 5.97953053592003115708021121908, 7.68355861699815976034773466784, 8.670232017103985326615636378902, 9.613172678273575213380080315908, 10.93556765869403135531574427146, 11.84566126881147538660492756759, 12.62930005829372530495542724517
