L(s) = 1 | + (0.866 + 0.5i)2-s + (0.965 + 0.258i)3-s + (0.499 + 0.866i)4-s + (−0.866 − 0.5i)5-s + (0.707 + 0.707i)6-s + (0.258 + 0.965i)7-s + 0.999i·8-s + (−0.499 − 0.866i)10-s + (0.965 + 0.258i)11-s + (0.258 + 0.965i)12-s + (−1 − i)13-s + (−0.258 + 0.965i)14-s + (−0.707 − 0.707i)15-s + (−0.5 + 0.866i)16-s + (0.965 − 0.258i)19-s − 0.999i·20-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (0.965 + 0.258i)3-s + (0.499 + 0.866i)4-s + (−0.866 − 0.5i)5-s + (0.707 + 0.707i)6-s + (0.258 + 0.965i)7-s + 0.999i·8-s + (−0.499 − 0.866i)10-s + (0.965 + 0.258i)11-s + (0.258 + 0.965i)12-s + (−1 − i)13-s + (−0.258 + 0.965i)14-s + (−0.707 − 0.707i)15-s + (−0.5 + 0.866i)16-s + (0.965 − 0.258i)19-s − 0.999i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.001556607\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.001556607\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 + (-0.258 - 0.965i)T \) |
| 41 | \( 1 + iT \) |
good | 3 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 5 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 13 | \( 1 + (1 + i)T + iT^{2} \) |
| 17 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.517 - 1.93i)T + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 83 | \( 1 + 1.41iT - T^{2} \) |
| 89 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.916969521385568394284194681178, −8.961795712815635977488654664503, −8.489496155986271900479460708879, −7.71839256315992173177156439929, −6.99540500425694596685785319420, −5.66169733007999891356421754748, −5.02521052375426968236993755004, −3.96418101350895169018833309470, −3.23674694464563908152235656260, −2.25003138689535089439492433312,
1.55654363773125349285127665573, 2.75887613751024940408010861768, 3.72620224288440561364614538965, 4.17401094041188191417683300717, 5.39693725031750229617200775356, 6.75923158685578083734765157756, 7.28199372394726754761749593199, 7.967128973885502635917156777253, 9.240421091908257635395805949254, 9.792376178107123052862478748365