| L(s) = 1 | + (−3.04 − 5.26i)5-s + (2.29 + 6.61i)7-s + (−13.9 − 8.04i)11-s + 14·13-s + (−6.08 + 10.5i)17-s + (−4.58 + 2.64i)19-s + (−27.8 + 16.0i)23-s + (−6 + 10.3i)25-s + 42.5·29-s + (20.6 + 11.9i)31-s + (27.8 − 32.1i)35-s + (−19 − 32.9i)37-s − 24.3·41-s + 74.0i·43-s + (−38.5 + 30.3i)49-s + ⋯ |
| L(s) = 1 | + (−0.608 − 1.05i)5-s + (0.327 + 0.944i)7-s + (−1.26 − 0.731i)11-s + 1.07·13-s + (−0.357 + 0.619i)17-s + (−0.241 + 0.139i)19-s + (−1.21 + 0.699i)23-s + (−0.239 + 0.415i)25-s + 1.46·29-s + (0.665 + 0.384i)31-s + (0.796 − 0.919i)35-s + (−0.513 − 0.889i)37-s − 0.593·41-s + 1.72i·43-s + (−0.785 + 0.618i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.250 - 0.968i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.250 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.058414468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.058414468\) |
| \(L(2)\) |
|
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 | \( 1 + (-2.29 - 6.61i)T \) |
| good | 5 | \( 1 + (3.04 + 5.26i)T + (-12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (13.9 + 8.04i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 14T + 169T^{2} \) |
| 17 | \( 1 + (6.08 - 10.5i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (4.58 - 2.64i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (27.8 - 16.0i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 42.5T + 841T^{2} \) |
| 31 | \( 1 + (-20.6 - 11.9i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (19 + 32.9i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 24.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 74.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (21.2 - 36.8i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-97.5 - 56.3i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-96.2 - 55.5i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 64.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-7 + 12.1i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (48.1 - 27.7i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 112. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-66.9 - 115. i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 35T + 9.40e3T^{2} \) |
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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
−9.948788429976281889318990050512, −8.660751482589139630367743776284, −8.458761042911547324166043570348, −7.85153094769542114708913792234, −6.31685108552146454015353143351, −5.59743036171589354218087697158, −4.76477060389022892880392826025, −3.76261232009564031272062349862, −2.51604287462831889780095290243, −1.12983099942067893053491458172,
0.37112899487998109896254800148, 2.15277323445666933722668235695, 3.26245686815758682467685982173, 4.20604671494607634376175620832, 5.09444138517933045639815224380, 6.52953023267606682352983673707, 6.98964578779768254674622261588, 7.966850804809102195304146289694, 8.428984459498076824703401385250, 10.04005501545889787370396553727
