| L(s) = 1 | + 7.05i·3-s + (9.69 + 5.57i)5-s + (−18.4 + 1.81i)7-s − 22.7·9-s + 22.9i·11-s + 35.9·13-s + (−39.3 + 68.3i)15-s − 80.0·17-s + 39.9·19-s + (−12.7 − 129. i)21-s − 183.·23-s + (62.8 + 108. i)25-s + 30.1i·27-s + 41.8·29-s − 223.·31-s + ⋯ |
| L(s) = 1 | + 1.35i·3-s + (0.866 + 0.498i)5-s + (−0.995 + 0.0978i)7-s − 0.841·9-s + 0.629i·11-s + 0.766·13-s + (−0.676 + 1.17i)15-s − 1.14·17-s + 0.481·19-s + (−0.132 − 1.35i)21-s − 1.66·23-s + (0.502 + 0.864i)25-s + 0.214i·27-s + 0.268·29-s − 1.29·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.039041675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.039041675\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-9.69 - 5.57i)T \) |
| 7 | \( 1 + (18.4 - 1.81i)T \) |
| good | 3 | \( 1 - 7.05iT - 27T^{2} \) |
| 11 | \( 1 - 22.9iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 35.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 80.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 39.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 183.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 41.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 223.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 91.3iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 138. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 169.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 92.1iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 258. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 250.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 65.9iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 658.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 370. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 511.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 26.3iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 1.12e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.59e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 927.T + 9.12e5T^{2} \) |
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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
−10.59563910769091796477319892739, −9.928123177384822854310743429426, −9.466026925652555221092016784635, −8.616870321696890676355572072736, −7.07447824550760892053165695142, −6.21698498497414548104465184384, −5.34827945321897764483970442898, −4.15062026329563984041484443045, −3.32286625231057977898920329352, −2.04836286001859435659108897722,
0.29348300921991138364711009147, 1.48932457252497201326492026421, 2.52705876382220522634493251787, 3.93273027880068922353824425131, 5.59231856389937948043835319734, 6.26208272916152084687547642804, 6.89564883910688665950149276960, 8.063300362731704918605986537448, 8.865483191515453063513871129246, 9.716027704822262142706530562699
