L(s) = 1 | + (0.631 − 0.631i)3-s + (2.34 + 2.34i)5-s − i·7-s + 2.20i·9-s + (−2.18 − 2.18i)11-s + (4.03 − 4.03i)13-s + 2.95·15-s + 0.347·17-s + (4.26 − 4.26i)19-s + (−0.631 − 0.631i)21-s + 6.23i·23-s + 5.97i·25-s + (3.28 + 3.28i)27-s + (−1.21 + 1.21i)29-s + 1.26·31-s + ⋯ |
L(s) = 1 | + (0.364 − 0.364i)3-s + (1.04 + 1.04i)5-s − 0.377i·7-s + 0.734i·9-s + (−0.658 − 0.658i)11-s + (1.11 − 1.11i)13-s + 0.763·15-s + 0.0843·17-s + (0.978 − 0.978i)19-s + (−0.137 − 0.137i)21-s + 1.30i·23-s + 1.19i·25-s + (0.632 + 0.632i)27-s + (−0.226 + 0.226i)29-s + 0.226·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.19678 + 0.114382i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.19678 + 0.114382i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (-0.631 + 0.631i)T - 3iT^{2} \) |
| 5 | \( 1 + (-2.34 - 2.34i)T + 5iT^{2} \) |
| 11 | \( 1 + (2.18 + 2.18i)T + 11iT^{2} \) |
| 13 | \( 1 + (-4.03 + 4.03i)T - 13iT^{2} \) |
| 17 | \( 1 - 0.347T + 17T^{2} \) |
| 19 | \( 1 + (-4.26 + 4.26i)T - 19iT^{2} \) |
| 23 | \( 1 - 6.23iT - 23T^{2} \) |
| 29 | \( 1 + (1.21 - 1.21i)T - 29iT^{2} \) |
| 31 | \( 1 - 1.26T + 31T^{2} \) |
| 37 | \( 1 + (-6.42 - 6.42i)T + 37iT^{2} \) |
| 41 | \( 1 - 2.68iT - 41T^{2} \) |
| 43 | \( 1 + (-4.05 - 4.05i)T + 43iT^{2} \) |
| 47 | \( 1 + 4.64T + 47T^{2} \) |
| 53 | \( 1 + (8.44 + 8.44i)T + 53iT^{2} \) |
| 59 | \( 1 + (5.17 + 5.17i)T + 59iT^{2} \) |
| 61 | \( 1 + (-0.00533 + 0.00533i)T - 61iT^{2} \) |
| 67 | \( 1 + (-3.02 + 3.02i)T - 67iT^{2} \) |
| 71 | \( 1 - 0.828iT - 71T^{2} \) |
| 73 | \( 1 - 6.25iT - 73T^{2} \) |
| 79 | \( 1 - 0.755T + 79T^{2} \) |
| 83 | \( 1 + (3.66 - 3.66i)T - 83iT^{2} \) |
| 89 | \( 1 + 6.24iT - 89T^{2} \) |
| 97 | \( 1 - 2.18T + 97T^{2} \) |
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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
−10.14031829135577518097525054407, −9.471356192864126975440512694454, −8.203762543386261352768276361077, −7.69499966946087164985548643513, −6.70054441590683815882349867372, −5.83652958541924072837546832543, −5.08006530027914274236332980091, −3.26396595290579944582701522718, −2.78692763475712642515944108396, −1.37812326826724319342326335904,
1.29185061519324759713733918141, 2.44368621687686454390564663175, 3.86929741722000825466258722866, 4.75053569800118212397672654240, 5.78765709674525399144604809817, 6.39373518570273113647032467354, 7.75550289073804827179279823085, 8.766552472614115056744451393797, 9.232527058992209017322270251216, 9.822211280903922276650103688904