| L(s) = 1 | + 2.79i·2-s − 1.73·3-s − 3.79·4-s − 4.83i·6-s + (5.63 − 4.15i)7-s + 0.578i·8-s + 2.99·9-s − 18.9·11-s + 6.56·12-s − 10.9·13-s + (11.6 + 15.7i)14-s − 16.7·16-s + 22.3·17-s + 8.37i·18-s − 19.6i·19-s + ⋯ |
| L(s) = 1 | + 1.39i·2-s − 0.577·3-s − 0.948·4-s − 0.805i·6-s + (0.804 − 0.593i)7-s + 0.0723i·8-s + 0.333·9-s − 1.72·11-s + 0.547·12-s − 0.844·13-s + (0.829 + 1.12i)14-s − 1.04·16-s + 1.31·17-s + 0.465i·18-s − 1.03i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6782116262\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6782116262\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 1.73T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-5.63 + 4.15i)T \) |
| good | 2 | \( 1 - 2.79iT - 4T^{2} \) |
| 11 | \( 1 + 18.9T + 121T^{2} \) |
| 13 | \( 1 + 10.9T + 169T^{2} \) |
| 17 | \( 1 - 22.3T + 289T^{2} \) |
| 19 | \( 1 + 19.6iT - 361T^{2} \) |
| 23 | \( 1 + 31.9iT - 529T^{2} \) |
| 29 | \( 1 + 39.9T + 841T^{2} \) |
| 31 | \( 1 + 36.6iT - 961T^{2} \) |
| 37 | \( 1 + 8.94iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 37.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 18.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 49.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 49.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 35.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 63.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 21.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 36.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + 6.66T + 5.32e3T^{2} \) |
| 79 | \( 1 - 16.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 36.7T + 6.88e3T^{2} \) |
| 89 | \( 1 + 88.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 133.T + 9.40e3T^{2} \) |
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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
−10.63083519969846669974444647430, −9.653412411070295402863793835963, −8.336438833041936836726838419546, −7.59278170023321425690110627628, −7.19204831379870187679324302958, −5.86453744984596604022029175031, −5.17989668203543023895308262610, −4.46994399547156725089211986256, −2.45586013013561470332622267601, −0.28498411856925217976112071645,
1.45208693836152642071911423043, 2.52676246937323071664091041728, 3.69503962753495696421837700846, 5.15929681309066561731605662236, 5.53643017575783278209266554279, 7.33597330513792977658569247743, 8.036857154617167416222605029847, 9.387692054019002097947731864106, 10.17460171747009157515862124319, 10.74084666996490245072681935379
