L(s) = 1 | + (−1 + 1.73i)2-s + (−1.99 − 3.46i)4-s + (−2.5 − 4.33i)5-s + (−7 + 12.1i)7-s + 7.99·8-s + 10·10-s + (3 − 5.19i)11-s + (−34 − 58.8i)13-s + (−14 − 24.2i)14-s + (−8 + 13.8i)16-s − 78·17-s + 44·19-s + (−10 + 17.3i)20-s + (6 + 10.3i)22-s + (60 + 103. i)23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s + (−0.377 + 0.654i)7-s + 0.353·8-s + 0.316·10-s + (0.0822 − 0.142i)11-s + (−0.725 − 1.25i)13-s + (−0.267 − 0.462i)14-s + (−0.125 + 0.216i)16-s − 1.11·17-s + 0.531·19-s + (−0.111 + 0.193i)20-s + (0.0581 + 0.100i)22-s + (0.543 + 0.942i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.011085024\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.011085024\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 - 1.73i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.5 + 4.33i)T \) |
good | 7 | \( 1 + (7 - 12.1i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-3 + 5.19i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (34 + 58.8i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 78T + 4.91e3T^{2} \) |
| 19 | \( 1 - 44T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-60 - 103. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-63 + 109. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-122 - 211. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 304T + 5.06e4T^{2} \) |
| 41 | \( 1 + (240 + 415. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (52 - 90.0i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-300 + 519. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 258T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-267 - 462. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (181 - 313. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-134 - 232. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 972T + 3.57e5T^{2} \) |
| 73 | \( 1 - 470T + 3.89e5T^{2} \) |
| 79 | \( 1 + (622 - 1.07e3i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-198 + 342. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 972T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-23 + 39.8i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−9.953484201120309725964394893902, −8.934941243252355244726807936089, −8.515850179315590759417505213659, −7.44341749514887886407294045500, −6.72567292619690724621766534315, −5.52153334075868838309731534712, −5.06998141274431255076505812897, −3.64036039217303153172313992457, −2.40494693836797730481215758588, −0.78096319082402285598252826912,
0.43419380193689987061901834077, 1.93059209658484318798592657299, 2.99504874152822162071524140063, 4.12057393426852355791066510729, 4.84456105884457559984360976997, 6.60112852050669371496419690328, 6.96378722362183589642910148184, 8.048376392844444442410170537612, 9.035196337251817847602833911429, 9.723226220786859470012486102080