L(s) = 1 | + (−1.12 − 0.301i)2-s + (−0.956 + 1.44i)3-s + (−0.555 − 0.320i)4-s + (−0.0307 − 0.114i)5-s + (1.51 − 1.33i)6-s + (−1.83 − 1.90i)7-s + (2.17 + 2.17i)8-s + (−1.16 − 2.76i)9-s + 0.138i·10-s + (2.30 + 2.30i)11-s + (0.995 − 0.495i)12-s + (2.91 − 2.12i)13-s + (1.48 + 2.70i)14-s + (0.194 + 0.0653i)15-s + (−1.15 − 1.99i)16-s + (0.665 − 1.15i)17-s + ⋯ |
L(s) = 1 | + (−0.795 − 0.213i)2-s + (−0.552 + 0.833i)3-s + (−0.277 − 0.160i)4-s + (−0.0137 − 0.0512i)5-s + (0.617 − 0.545i)6-s + (−0.692 − 0.721i)7-s + (0.769 + 0.769i)8-s + (−0.389 − 0.920i)9-s + 0.0437i·10-s + (0.695 + 0.695i)11-s + (0.287 − 0.143i)12-s + (0.808 − 0.589i)13-s + (0.397 + 0.722i)14-s + (0.0503 + 0.0168i)15-s + (−0.288 − 0.498i)16-s + (0.161 − 0.279i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.612622 - 0.0456210i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.612622 - 0.0456210i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.956 - 1.44i)T \) |
| 7 | \( 1 + (1.83 + 1.90i)T \) |
| 13 | \( 1 + (-2.91 + 2.12i)T \) |
good | 2 | \( 1 + (1.12 + 0.301i)T + (1.73 + i)T^{2} \) |
| 5 | \( 1 + (0.0307 + 0.114i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-2.30 - 2.30i)T + 11iT^{2} \) |
| 17 | \( 1 + (-0.665 + 1.15i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.29 - 3.29i)T + 19iT^{2} \) |
| 23 | \( 1 + (-1.93 - 3.34i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.77 - 2.75i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.08 + 7.79i)T + (-26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (-1.91 - 0.512i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (3.06 + 11.4i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (0.721 - 0.416i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (6.90 - 1.85i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (5.67 - 3.27i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.47 + 1.19i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 - 8.26T + 61T^{2} \) |
| 67 | \( 1 + (0.646 + 0.646i)T + 67iT^{2} \) |
| 71 | \( 1 + (-15.9 - 4.26i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (12.6 + 3.38i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-1.52 + 2.64i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.83 - 2.83i)T - 83iT^{2} \) |
| 89 | \( 1 + (1.62 - 6.05i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-3.43 + 12.8i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−11.54019545815018559124053401683, −10.64281914402703671888322755646, −9.923742578661035039156689651747, −9.400229123972569200568172344136, −8.310351105394360010973760522769, −6.99638387418885343566704329109, −5.76505355222807521093180658776, −4.59657841697579108034127734936, −3.49709308175194693248198796064, −0.929861588861227079151759657761,
1.09063437366182373252747475895, 3.18432306891215279991246160962, 4.92471563108568361250550713620, 6.38968175386796842410675151208, 6.84566895550873805081337203026, 8.307554669546901856668155664784, 8.800135007997367822940122241922, 9.854195337634118167877701210199, 11.06428798074744591019912612892, 11.87039412778382291087567730494