L(s) = 1 | + (1.65 − 1.65i)2-s + (0.818 − 1.52i)3-s − 3.46i·4-s + (−1.96 + 1.96i)5-s + (−1.17 − 3.87i)6-s + (0.707 − 0.707i)7-s + (−2.42 − 2.42i)8-s + (−1.66 − 2.49i)9-s + 6.48i·10-s + (−2.17 − 2.17i)11-s + (−5.28 − 2.83i)12-s + (2.86 + 2.18i)13-s − 2.33i·14-s + (1.39 + 4.60i)15-s − 1.07·16-s + 3.32·17-s + ⋯ |
L(s) = 1 | + (1.16 − 1.16i)2-s + (0.472 − 0.881i)3-s − 1.73i·4-s + (−0.877 + 0.877i)5-s + (−0.478 − 1.58i)6-s + (0.267 − 0.267i)7-s + (−0.855 − 0.855i)8-s + (−0.553 − 0.832i)9-s + 2.05i·10-s + (−0.655 − 0.655i)11-s + (−1.52 − 0.818i)12-s + (0.795 + 0.605i)13-s − 0.624i·14-s + (0.359 + 1.18i)15-s − 0.268·16-s + 0.807·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.02637 - 2.07198i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.02637 - 2.07198i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.818 + 1.52i)T \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
| 13 | \( 1 + (-2.86 - 2.18i)T \) |
good | 2 | \( 1 + (-1.65 + 1.65i)T - 2iT^{2} \) |
| 5 | \( 1 + (1.96 - 1.96i)T - 5iT^{2} \) |
| 11 | \( 1 + (2.17 + 2.17i)T + 11iT^{2} \) |
| 17 | \( 1 - 3.32T + 17T^{2} \) |
| 19 | \( 1 + (-2.91 - 2.91i)T + 19iT^{2} \) |
| 23 | \( 1 - 0.190T + 23T^{2} \) |
| 29 | \( 1 - 7.09iT - 29T^{2} \) |
| 31 | \( 1 + (-1.90 - 1.90i)T + 31iT^{2} \) |
| 37 | \( 1 + (6.97 - 6.97i)T - 37iT^{2} \) |
| 41 | \( 1 + (6.16 - 6.16i)T - 41iT^{2} \) |
| 43 | \( 1 + 12.4iT - 43T^{2} \) |
| 47 | \( 1 + (2.28 + 2.28i)T + 47iT^{2} \) |
| 53 | \( 1 + 5.76iT - 53T^{2} \) |
| 59 | \( 1 + (2.95 + 2.95i)T + 59iT^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + (4.16 + 4.16i)T + 67iT^{2} \) |
| 71 | \( 1 + (1.60 - 1.60i)T - 71iT^{2} \) |
| 73 | \( 1 + (-0.722 + 0.722i)T - 73iT^{2} \) |
| 79 | \( 1 - 7.14T + 79T^{2} \) |
| 83 | \( 1 + (3.15 - 3.15i)T - 83iT^{2} \) |
| 89 | \( 1 + (-5.99 - 5.99i)T + 89iT^{2} \) |
| 97 | \( 1 + (7.27 + 7.27i)T + 97iT^{2} \) |
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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
−11.77565085713233705562899695013, −11.01513144315781451433080753651, −10.20994806388404389308730890093, −8.539435738385662988683473418817, −7.59309384574878469167750568170, −6.49932693883514855694232841862, −5.21683147500454393544211206699, −3.50728367230084205699213823347, −3.23359407351902953787887923298, −1.53803947017575477522547357829,
3.17098392498293021533511343572, 4.27657638757524634102378106796, 4.97862219208454771740269442283, 5.83238109079960727010574539432, 7.60348244838549565218501638943, 8.013441220676467713671198912019, 9.024894489643429807406549299983, 10.31961693226683662605879170625, 11.61833137222244853779595767508, 12.50338662528422617874632477335