L(s) = 1 | + (1.5 − 0.866i)2-s + (0.5 − 0.866i)4-s − 1.73i·5-s + 1.73i·8-s + (−1.49 − 2.59i)10-s + (−2.5 + 2.59i)13-s + (2.49 + 4.33i)16-s + (−1.5 + 2.59i)17-s + (−3 − 1.73i)19-s + (−1.50 − 0.866i)20-s + (−3 − 5.19i)23-s + 2.00·25-s + (−1.5 + 6.06i)26-s + (1.5 + 2.59i)29-s − 3.46i·31-s + (4.5 + 2.59i)32-s + ⋯ |
L(s) = 1 | + (1.06 − 0.612i)2-s + (0.250 − 0.433i)4-s − 0.774i·5-s + 0.612i·8-s + (−0.474 − 0.821i)10-s + (−0.693 + 0.720i)13-s + (0.624 + 1.08i)16-s + (−0.363 + 0.630i)17-s + (−0.688 − 0.397i)19-s + (−0.335 − 0.193i)20-s + (−0.625 − 1.08i)23-s + 0.400·25-s + (−0.294 + 1.18i)26-s + (0.278 + 0.482i)29-s − 0.622i·31-s + (0.795 + 0.459i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.50165 - 0.616372i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.50165 - 0.616372i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 + (2.5 - 2.59i)T \) |
good | 2 | \( 1 + (-1.5 + 0.866i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 7 | \( 1 + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + (-7.5 + 4.33i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.5 + 2.59i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 3.46iT - 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + (6 + 3.46i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3 + 1.73i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3 + 1.73i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 1.73iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 13.8iT - 83T^{2} \) |
| 89 | \( 1 + (-6 + 3.46i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6 - 3.46i)T + (48.5 + 84.0i)T^{2} \) |
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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
−13.09663201674913409202885206212, −12.58913598375338648718720352897, −11.65594467670936743887824648803, −10.60685065900241612146680633728, −9.147738395607172997143644102893, −8.121361817311082119247377055501, −6.38007784589126407921637885043, −4.93028819176766711137701447128, −4.12193749243753436827426393249, −2.30048864547383042155892541730,
3.01061525879386457489213948438, 4.49171745133503383136198757094, 5.75306601814164473145468760627, 6.79240429228524573602732606305, 7.81619518739241434056211690148, 9.578556319675191966040292762700, 10.56540634359934074886604029254, 11.87492882753469792665319424840, 12.91113829320065756769769732237, 13.82537752038117811414386352950