L(s) = 1 | + (−1.22 + 1.22i)2-s + (1.22 + 1.22i)3-s − 0.999i·4-s − 2.99·6-s + (−1.22 − 1.22i)8-s + 2.99i·9-s + (1.22 − 1.22i)12-s + 5·16-s + (4.89 − 4.89i)17-s + (−3.67 − 3.67i)18-s − 4i·19-s + (2.44 + 2.44i)23-s − 3.00i·24-s + (−3.67 + 3.67i)27-s − 8·31-s + (−3.67 + 3.67i)32-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.866i)2-s + (0.707 + 0.707i)3-s − 0.499i·4-s − 1.22·6-s + (−0.433 − 0.433i)8-s + 0.999i·9-s + (0.353 − 0.353i)12-s + 1.25·16-s + (1.18 − 1.18i)17-s + (−0.866 − 0.866i)18-s − 0.917i·19-s + (0.510 + 0.510i)23-s − 0.612i·24-s + (−0.707 + 0.707i)27-s − 1.43·31-s + (−0.649 + 0.649i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.464074 + 0.586383i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.464074 + 0.586383i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.22 - 1.22i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (1.22 - 1.22i)T - 2iT^{2} \) |
| 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + (-4.89 + 4.89i)T - 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (-2.44 - 2.44i)T + 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + (7.34 - 7.34i)T - 47iT^{2} \) |
| 53 | \( 1 + (9.79 + 9.79i)T + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 73iT^{2} \) |
| 79 | \( 1 - 16iT - 79T^{2} \) |
| 83 | \( 1 + (-2.44 - 2.44i)T + 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97iT^{2} \) |
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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
−15.09923938371301463899068690834, −14.20182649593418830304353997041, −12.89688620801135847542192496650, −11.29016462855030472015195467614, −9.819690080857000099844974709186, −9.186000810889273437221384996387, −8.030321890646285451268884181389, −7.07167264269865822101784048619, −5.23175479693293312759926761244, −3.28819910129529287773214584801,
1.64009472245150218811766256515, 3.34831099937780030944362603971, 5.97508446664511383909886609544, 7.69629943029747480888817969025, 8.635593233161787413572018965313, 9.709095682617365801481477083698, 10.76081486330043137405137605251, 12.09796814800089766133138762862, 12.81805657091748060312427666734, 14.32271057770778321424788961131