L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.226 − 2.22i)5-s + (−2.57 − 2.57i)7-s + (−0.707 − 0.707i)8-s + (−1.41 − 1.73i)10-s − 4.44i·11-s + (2 − 2i)13-s − 3.64·14-s − 1.00·16-s + (−4.09 + 4.09i)17-s + 0.825i·19-s + (−2.22 − 0.226i)20-s + (−3.14 − 3.14i)22-s + (0.707 + 0.707i)23-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.101 − 0.994i)5-s + (−0.974 − 0.974i)7-s + (−0.250 − 0.250i)8-s + (−0.446 − 0.548i)10-s − 1.34i·11-s + (0.554 − 0.554i)13-s − 0.974·14-s − 0.250·16-s + (−0.993 + 0.993i)17-s + 0.189i·19-s + (−0.497 − 0.0506i)20-s + (−0.670 − 0.670i)22-s + (0.147 + 0.147i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.482i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.876 - 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.376780565\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.376780565\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.226 + 2.22i)T \) |
| 23 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 + (2.57 + 2.57i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.44iT - 11T^{2} \) |
| 13 | \( 1 + (-2 + 2i)T - 13iT^{2} \) |
| 17 | \( 1 + (4.09 - 4.09i)T - 17iT^{2} \) |
| 19 | \( 1 - 0.825iT - 19T^{2} \) |
| 29 | \( 1 + 1.25T + 29T^{2} \) |
| 31 | \( 1 - 5.79T + 31T^{2} \) |
| 37 | \( 1 + (-1.43 - 1.43i)T + 37iT^{2} \) |
| 41 | \( 1 - 3.57iT - 41T^{2} \) |
| 43 | \( 1 + (0.569 - 0.569i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.86 + 1.86i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.80 + 2.80i)T + 53iT^{2} \) |
| 59 | \( 1 - 5.30T + 59T^{2} \) |
| 61 | \( 1 - 0.328T + 61T^{2} \) |
| 67 | \( 1 + (-9.53 - 9.53i)T + 67iT^{2} \) |
| 71 | \( 1 + 10.6iT - 71T^{2} \) |
| 73 | \( 1 + (3.90 - 3.90i)T - 73iT^{2} \) |
| 79 | \( 1 - 15.8iT - 79T^{2} \) |
| 83 | \( 1 + (9.36 + 9.36i)T + 83iT^{2} \) |
| 89 | \( 1 + 8.63T + 89T^{2} \) |
| 97 | \( 1 + (6.39 + 6.39i)T + 97iT^{2} \) |
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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
−8.576206375314616742226666051042, −8.215707823005372876628493082246, −6.86589556620385907978861171250, −6.12074262800003524542811100499, −5.51442467181313762968553108192, −4.33553265095779314250066539483, −3.77264091140911819693814784169, −2.89343690451179909381174927725, −1.37586200715572029471966355861, −0.41036324393651775957288165548,
2.22368749872433721498406029738, 2.78871377290056786500308559356, 3.87830330316636307854560029846, 4.77752089709181429749624975780, 5.76637398239296316566088884869, 6.64543018747714423336248221063, 6.83692701555841872623778460959, 7.78395658047843660669816028007, 8.941599803262205758915988399162, 9.424821468810879801989478464592