| L(s) = 1 | + (−1 − i)3-s + (−1 − 2i)5-s − 2i·7-s − i·9-s + (1 − i)11-s + (3 − 2i)13-s + (−1 + 3i)15-s + (1 + i)17-s + (5 − 5i)19-s + (−2 + 2i)21-s + (−3 + 3i)23-s + (−3 + 4i)25-s + (−4 + 4i)27-s + (−5 − 5i)31-s − 2·33-s + ⋯ |
| L(s) = 1 | + (−0.577 − 0.577i)3-s + (−0.447 − 0.894i)5-s − 0.755i·7-s − 0.333i·9-s + (0.301 − 0.301i)11-s + (0.832 − 0.554i)13-s + (−0.258 + 0.774i)15-s + (0.242 + 0.242i)17-s + (1.14 − 1.14i)19-s + (−0.436 + 0.436i)21-s + (−0.625 + 0.625i)23-s + (−0.600 + 0.800i)25-s + (−0.769 + 0.769i)27-s + (−0.898 − 0.898i)31-s − 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9974126837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9974126837\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1 + 2i)T \) |
| 13 | \( 1 + (-3 + 2i)T \) |
| good | 3 | \( 1 + (1 + i)T + 3iT^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + (-1 + i)T - 11iT^{2} \) |
| 17 | \( 1 + (-1 - i)T + 17iT^{2} \) |
| 19 | \( 1 + (-5 + 5i)T - 19iT^{2} \) |
| 23 | \( 1 + (3 - 3i)T - 23iT^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + (5 + 5i)T + 31iT^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + (7 + 7i)T + 41iT^{2} \) |
| 43 | \( 1 + (-1 + i)T - 43iT^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + (-5 - 5i)T + 53iT^{2} \) |
| 59 | \( 1 + (-7 - 7i)T + 59iT^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + (1 + i)T + 71iT^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 2iT - 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + (-5 - 5i)T + 89iT^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.381860490108340154208129043481, −8.765315648582686084653789601722, −7.65537078506029697498044668597, −7.19099344386093477252575494177, −6.00425955772248723458492923333, −5.40180556738381697636783232904, −4.14949931030849773213435971222, −3.39367195697155453233074903669, −1.38007376366451335470087447811, −0.53293961281078586966922446857,
1.88397121128998389256363157172, 3.24633093681171605971402491942, 4.10572335095715121257375310446, 5.22183404933944131072661669707, 5.98867042160851162451703799398, 6.86140352120462323526462552461, 7.83100910378069500059522272506, 8.643512766032552506770584951776, 9.742245056173822241157573003631, 10.30869352353359632981491321057
