L(s) = 1 | + (1 − i)2-s + (1.22 + 1.22i)3-s − 2i·4-s + (−2.41 + 4.37i)5-s + 2.44·6-s + (0.381 − 0.381i)7-s + (−2 − 2i)8-s + 2.99i·9-s + (1.96 + 6.79i)10-s + 6.70·11-s + (2.44 − 2.44i)12-s + (3.27 + 3.27i)13-s − 0.763i·14-s + (−8.32 + 2.40i)15-s − 4·16-s + (−12.9 + 12.9i)17-s + ⋯ |
L(s) = 1 | + (0.5 − 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s + (−0.483 + 0.875i)5-s + 0.408·6-s + (0.0545 − 0.0545i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (0.196 + 0.679i)10-s + 0.609·11-s + (0.204 − 0.204i)12-s + (0.252 + 0.252i)13-s − 0.0545i·14-s + (−0.554 + 0.160i)15-s − 0.250·16-s + (−0.763 + 0.763i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.269 - 0.963i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.269 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.129854990\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.129854990\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + i)T \) |
| 3 | \( 1 + (-1.22 - 1.22i)T \) |
| 5 | \( 1 + (2.41 - 4.37i)T \) |
| 23 | \( 1 + (3.39 + 3.39i)T \) |
good | 7 | \( 1 + (-0.381 + 0.381i)T - 49iT^{2} \) |
| 11 | \( 1 - 6.70T + 121T^{2} \) |
| 13 | \( 1 + (-3.27 - 3.27i)T + 169iT^{2} \) |
| 17 | \( 1 + (12.9 - 12.9i)T - 289iT^{2} \) |
| 19 | \( 1 - 18.6iT - 361T^{2} \) |
| 29 | \( 1 - 43.2iT - 841T^{2} \) |
| 31 | \( 1 - 14.8T + 961T^{2} \) |
| 37 | \( 1 + (7.77 - 7.77i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 19.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-41.2 - 41.2i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (31.7 - 31.7i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-13.9 - 13.9i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 2.31iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 22.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-59.1 + 59.1i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 29.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + (73.2 + 73.2i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 12.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (12.8 + 12.8i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 42.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-8.47 + 8.47i)T - 9.40e3iT^{2} \) |
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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
−10.64894255666613333403394393680, −9.775098727143484340476287726867, −8.810442026469355308137136450131, −7.902972077505592143132458198569, −6.76970167687003086499181007322, −6.01532255383786253102773528041, −4.57968506516516199250919214222, −3.82149979274264959411226488120, −2.98987568575545062288715091141, −1.67496897677491803685133689782,
0.61206761322573380914735252725, 2.30060273423773960585078719590, 3.68553232715157024260457947864, 4.54430350403605956883456410486, 5.51851537002875786337392299638, 6.65843532500349196162252568122, 7.39121926813497234630126413025, 8.390841784629057977130008063536, 8.880769012506943147136815202426, 9.827597541085834019093590750267