L(s) = 1 | + (0.884 − 0.705i)3-s + (0.449 − 0.216i)5-s + (2.01 + 2.52i)7-s + (−0.382 + 1.67i)9-s + (2.96 − 0.676i)11-s + (−1.36 − 5.99i)13-s + (0.244 − 0.508i)15-s − 2.56i·17-s + (1.46 + 1.17i)19-s + (3.55 + 0.811i)21-s + (−3.94 − 1.90i)23-s + (−2.96 + 3.71i)25-s + (2.31 + 4.81i)27-s + (−5.30 − 0.904i)29-s + (2.42 + 5.04i)31-s + ⋯ |
L(s) = 1 | + (0.510 − 0.407i)3-s + (0.200 − 0.0967i)5-s + (0.759 + 0.952i)7-s + (−0.127 + 0.558i)9-s + (0.893 − 0.204i)11-s + (−0.379 − 1.66i)13-s + (0.0632 − 0.131i)15-s − 0.623i·17-s + (0.336 + 0.268i)19-s + (0.776 + 0.177i)21-s + (−0.823 − 0.396i)23-s + (−0.592 + 0.742i)25-s + (0.445 + 0.925i)27-s + (−0.985 − 0.167i)29-s + (0.435 + 0.905i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.55111 - 0.124756i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.55111 - 0.124756i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 + (5.30 + 0.904i)T \) |
good | 3 | \( 1 + (-0.884 + 0.705i)T + (0.667 - 2.92i)T^{2} \) |
| 5 | \( 1 + (-0.449 + 0.216i)T + (3.11 - 3.90i)T^{2} \) |
| 7 | \( 1 + (-2.01 - 2.52i)T + (-1.55 + 6.82i)T^{2} \) |
| 11 | \( 1 + (-2.96 + 0.676i)T + (9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (1.36 + 5.99i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + 2.56iT - 17T^{2} \) |
| 19 | \( 1 + (-1.46 - 1.17i)T + (4.22 + 18.5i)T^{2} \) |
| 23 | \( 1 + (3.94 + 1.90i)T + (14.3 + 17.9i)T^{2} \) |
| 31 | \( 1 + (-2.42 - 5.04i)T + (-19.3 + 24.2i)T^{2} \) |
| 37 | \( 1 + (2.36 + 0.539i)T + (33.3 + 16.0i)T^{2} \) |
| 41 | \( 1 + 6.10iT - 41T^{2} \) |
| 43 | \( 1 + (2.83 - 5.87i)T + (-26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (6.57 - 1.50i)T + (42.3 - 20.3i)T^{2} \) |
| 53 | \( 1 + (0.169 - 0.0814i)T + (33.0 - 41.4i)T^{2} \) |
| 59 | \( 1 - 6.11T + 59T^{2} \) |
| 61 | \( 1 + (8.61 - 6.86i)T + (13.5 - 59.4i)T^{2} \) |
| 67 | \( 1 + (-2.07 + 9.10i)T + (-60.3 - 29.0i)T^{2} \) |
| 71 | \( 1 + (1.82 + 7.97i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (2.47 - 5.13i)T + (-45.5 - 57.0i)T^{2} \) |
| 79 | \( 1 + (14.1 + 3.22i)T + (71.1 + 34.2i)T^{2} \) |
| 83 | \( 1 + (-4.09 + 5.13i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-2.74 - 5.68i)T + (-55.4 + 69.5i)T^{2} \) |
| 97 | \( 1 + (-6.19 - 4.94i)T + (21.5 + 94.5i)T^{2} \) |
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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
−12.18151134869413444904577337725, −11.36221517895357248205914753998, −10.19771280128098979237868140403, −9.046836988853705246683278086083, −8.203739924204663239736305050125, −7.44600963440556148800569518179, −5.85065557209399936688757008064, −5.02010367166875107570315714749, −3.15002390301082293649843890208, −1.83857497946936598121793586873,
1.80877587690735655103644500600, 3.80252315329595517663666604507, 4.45117730260420698615712406115, 6.22930531150486936931883950706, 7.20875518058562840460368976612, 8.389008258943067162326660242505, 9.421340950278514724917883292902, 10.04840167368236693459238753657, 11.42279350185255501900547067286, 11.91596355420528741164927789331