L(s) = 1 | + (7.85 − 4.53i)5-s + (−1.17 − 6.90i)7-s + (8.81 − 15.2i)11-s + 19.0i·13-s + (14.7 + 8.53i)17-s + (7.74 − 4.46i)19-s + (−4.72 − 8.18i)23-s + (28.6 − 49.6i)25-s + 10.0·29-s + (39.5 + 22.8i)31-s + (−40.5 − 48.8i)35-s + (−26.9 − 46.6i)37-s + 58.9i·41-s − 69.3·43-s + (−41.7 + 24.0i)47-s + ⋯ |
L(s) = 1 | + (1.57 − 0.907i)5-s + (−0.168 − 0.985i)7-s + (0.801 − 1.38i)11-s + 1.46i·13-s + (0.869 + 0.501i)17-s + (0.407 − 0.235i)19-s + (−0.205 − 0.355i)23-s + (1.14 − 1.98i)25-s + 0.345·29-s + (1.27 + 0.736i)31-s + (−1.15 − 1.39i)35-s + (−0.727 − 1.26i)37-s + 1.43i·41-s − 1.61·43-s + (−0.887 + 0.512i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.291 + 0.956i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.291 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.757011865\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.757011865\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.17 + 6.90i)T \) |
good | 5 | \( 1 + (-7.85 + 4.53i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-8.81 + 15.2i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 19.0iT - 169T^{2} \) |
| 17 | \( 1 + (-14.7 - 8.53i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-7.74 + 4.46i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (4.72 + 8.18i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 10.0T + 841T^{2} \) |
| 31 | \( 1 + (-39.5 - 22.8i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (26.9 + 46.6i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 58.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 69.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + (41.7 - 24.0i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-21.6 + 37.4i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (0.930 + 0.537i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-45.3 + 26.2i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-24.0 + 41.5i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 53.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + (2.16 + 1.25i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-11.2 - 19.5i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 51.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (43.9 - 25.3i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 126. iT - 9.40e3T^{2} \) |
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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
−9.673910313535724545590375728343, −8.842354150947217691201340907565, −8.212872022683486845952785647459, −6.67536985794516179704411994729, −6.33644477515115451855018521560, −5.29104222653062787220872981921, −4.37494689306761562533020757932, −3.26477991405325750925578892799, −1.69376027832511673763876565562, −0.944534369693899586767846435702,
1.51697583352488770380053080603, 2.52152740057939772133495923815, 3.30803251892427349773662386334, 5.04945019568861391947389524093, 5.66541684188278478336873579243, 6.47087171091422355963504641566, 7.23176149434277044503512062761, 8.349034882643381593548877950267, 9.420928078023117434275590433340, 10.06170846653687731631652692532