L(s) = 1 | + (10.9 + 18.9i)5-s + (−12.1 − 13.9i)7-s + (45.1 + 26.0i)11-s − 54.9i·13-s + (40.8 − 70.7i)17-s + (113. − 65.6i)19-s + (38.0 − 21.9i)23-s + (−176. + 305. i)25-s + 238. i·29-s + (174. + 100. i)31-s + (132. − 382. i)35-s + (12.0 + 20.8i)37-s − 102.·41-s + 119.·43-s + (20.2 + 35.1i)47-s + ⋯ |
L(s) = 1 | + (0.977 + 1.69i)5-s + (−0.655 − 0.755i)7-s + (1.23 + 0.714i)11-s − 1.17i·13-s + (0.582 − 1.00i)17-s + (1.37 − 0.792i)19-s + (0.345 − 0.199i)23-s + (−1.40 + 2.44i)25-s + 1.52i·29-s + (1.00 + 0.582i)31-s + (0.638 − 1.84i)35-s + (0.0534 + 0.0925i)37-s − 0.389·41-s + 0.424·43-s + (0.0629 + 0.109i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 - 0.681i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.732 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.481108682\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.481108682\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (12.1 + 13.9i)T \) |
good | 5 | \( 1 + (-10.9 - 18.9i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-45.1 - 26.0i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 54.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-40.8 + 70.7i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-113. + 65.6i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-38.0 + 21.9i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 238. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-174. - 100. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-12.0 - 20.8i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 102.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 119.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-20.2 - 35.1i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (297. + 172. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (142. - 246. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (386. - 223. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-113. + 196. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 886. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-6.46 - 3.73i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (404. + 700. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 943.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-575. - 996. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.55e3iT - 9.12e5T^{2} \) |
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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
−10.45358969241387075927557168030, −9.851534573691085301255418668961, −9.236454371300399600766813682580, −7.39617861455696987256319303738, −7.02876922863348840712190475299, −6.21996866374522581598894977607, −5.05925354691474937589084777628, −3.35887229622870502519699766555, −2.87735534851631908914452962201, −1.15934666237594962032431588626,
0.967171093570407781296498826268, 1.92339431732729491102640417069, 3.64541472073819888896870161400, 4.76147096571807043553279909994, 6.02688030662826578349693869900, 6.13062221356758089370864844913, 7.979304895337146450228994143505, 8.885293534400893959829616448670, 9.407456179240836804766528945881, 9.959903474789717179950388380935