L(s) = 1 | + (0.707 − 0.707i)2-s + (−1.41 − 1.41i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s − 2.00·6-s + (−1 + i)7-s + (−0.707 − 0.707i)8-s + 3.00i·9-s + 1.00·10-s + (−1.41 + 1.41i)12-s + 1.41i·14-s − 2.00i·15-s − 1.00·16-s + (−1 + i)17-s + (2.12 + 2.12i)18-s + 1.41i·19-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s + (−1.41 − 1.41i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s − 2.00·6-s + (−1 + i)7-s + (−0.707 − 0.707i)8-s + 3.00i·9-s + 1.00·10-s + (−1.41 + 1.41i)12-s + 1.41i·14-s − 2.00i·15-s − 1.00·16-s + (−1 + i)17-s + (2.12 + 2.12i)18-s + 1.41i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6310379458\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6310379458\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 53 | \( 1 + (0.707 + 0.707i)T \) |
good | 3 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 7 | \( 1 + (1 - i)T - iT^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (1 - i)T - iT^{2} \) |
| 19 | \( 1 - 1.41iT - T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (1 - i)T - iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + 1.41T + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-1 + i)T - iT^{2} \) |
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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
−9.689167431289596481125318229390, −8.529679084551758797084930412608, −7.35929341212984009466342250237, −6.36255438184508091917994091035, −6.20568999335461215660493781035, −5.70608221520693408555721347467, −4.72344808314074669415318623964, −3.24050446981285120949565517736, −2.20759381300570356902980962090, −1.62850064485113365967521318389,
0.40124911765435560377601349650, 2.97408879401533061341151497796, 3.93438864959508059739449142373, 4.75312624334806365267076648480, 5.00322436537271352260514757553, 6.08521568343809063152214536181, 6.54429670636421316104179274616, 7.21109097804907032615542183665, 8.849830878676702770400700631184, 9.290312192840133844527898497701