| L(s) = 1 | + (4.76 − 5.68i)2-s + (1.42 + 3.91i)3-s + (−6.78 − 38.4i)4-s + (−7.62 + 43.2i)5-s + (29.0 + 10.5i)6-s + (−9.09 − 15.7i)7-s + (−148. − 85.5i)8-s + (48.7 − 40.8i)9-s + (209. + 249. i)10-s + (−47.7 + 82.7i)11-s + (140. − 81.3i)12-s + (14.7 − 40.5i)13-s + (−132. − 23.4i)14-s + (−180. + 31.7i)15-s + (−605. + 220. i)16-s + (58.4 + 49.0i)17-s + ⋯ |
| L(s) = 1 | + (1.19 − 1.42i)2-s + (0.158 + 0.435i)3-s + (−0.423 − 2.40i)4-s + (−0.304 + 1.72i)5-s + (0.807 + 0.293i)6-s + (−0.185 − 0.321i)7-s + (−2.31 − 1.33i)8-s + (0.601 − 0.504i)9-s + (2.09 + 2.49i)10-s + (−0.394 + 0.684i)11-s + (0.978 − 0.565i)12-s + (0.0873 − 0.240i)13-s + (−0.678 − 0.119i)14-s + (−0.800 + 0.141i)15-s + (−2.36 + 0.860i)16-s + (0.202 + 0.169i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.64751 - 1.16976i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64751 - 1.16976i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (310. + 184. i)T \) |
| good | 2 | \( 1 + (-4.76 + 5.68i)T + (-2.77 - 15.7i)T^{2} \) |
| 3 | \( 1 + (-1.42 - 3.91i)T + (-62.0 + 52.0i)T^{2} \) |
| 5 | \( 1 + (7.62 - 43.2i)T + (-587. - 213. i)T^{2} \) |
| 7 | \( 1 + (9.09 + 15.7i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (47.7 - 82.7i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-14.7 + 40.5i)T + (-2.18e4 - 1.83e4i)T^{2} \) |
| 17 | \( 1 + (-58.4 - 49.0i)T + (1.45e4 + 8.22e4i)T^{2} \) |
| 23 | \( 1 + (69.1 + 392. i)T + (-2.62e5 + 9.57e4i)T^{2} \) |
| 29 | \( 1 + (-323. - 385. i)T + (-1.22e5 + 6.96e5i)T^{2} \) |
| 31 | \( 1 + (-788. + 455. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 - 2.20e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-207. - 571. i)T + (-2.16e6 + 1.81e6i)T^{2} \) |
| 43 | \( 1 + (9.54 - 54.1i)T + (-3.21e6 - 1.16e6i)T^{2} \) |
| 47 | \( 1 + (-1.77e3 + 1.48e3i)T + (8.47e5 - 4.80e6i)T^{2} \) |
| 53 | \( 1 + (-445. + 78.6i)T + (7.41e6 - 2.69e6i)T^{2} \) |
| 59 | \( 1 + (867. - 1.03e3i)T + (-2.10e6 - 1.19e7i)T^{2} \) |
| 61 | \( 1 + (-778. - 4.41e3i)T + (-1.30e7 + 4.73e6i)T^{2} \) |
| 67 | \( 1 + (4.19e3 + 5.00e3i)T + (-3.49e6 + 1.98e7i)T^{2} \) |
| 71 | \( 1 + (959. + 169. i)T + (2.38e7 + 8.69e6i)T^{2} \) |
| 73 | \( 1 + (5.97e3 - 2.17e3i)T + (2.17e7 - 1.82e7i)T^{2} \) |
| 79 | \( 1 + (-103. - 284. i)T + (-2.98e7 + 2.50e7i)T^{2} \) |
| 83 | \( 1 + (-1.10e3 - 1.91e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (2.77e3 - 7.62e3i)T + (-4.80e7 - 4.03e7i)T^{2} \) |
| 97 | \( 1 + (4.94e3 - 5.89e3i)T + (-1.53e7 - 8.71e7i)T^{2} \) |
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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
−18.17939503428130341990493639080, −15.27449148789898365011698081663, −14.80546296077955349976451182849, −13.43736479390951571023026685190, −12.04050531205743607231536365422, −10.61460875141709063802468389478, −10.09540308939348722115858855514, −6.64216557599998440675814420982, −4.21547709342181321675451350484, −2.82866089810244274067481817117,
4.38138855671502910713560099629, 5.74747791742276902015115805349, 7.69110104020813955836194718566, 8.708925364612333989644694575360, 12.22708611771310611589392779731, 12.97850575244870793539343728440, 13.91225770830960699125426984011, 15.73611568230295781651384344876, 16.21939189425374430870397259045, 17.34080533248166841303304545670
