| L(s) = 1 | + (−0.347 + 1.96i)2-s + (3.93 − 3.38i)3-s + (−3.75 − 1.36i)4-s + (7.73 − 6.49i)5-s + (5.30 + 8.93i)6-s + (7.54 − 2.74i)7-s + (4 − 6.92i)8-s + (4.02 − 26.6i)9-s + (10.1 + 17.4i)10-s + (12.9 + 10.8i)11-s + (−19.4 + 7.35i)12-s + (14.9 + 84.8i)13-s + (2.79 + 15.8i)14-s + (8.47 − 51.8i)15-s + (12.2 + 10.2i)16-s + (−53.3 − 92.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (0.757 − 0.652i)3-s + (−0.469 − 0.171i)4-s + (0.692 − 0.580i)5-s + (0.361 + 0.607i)6-s + (0.407 − 0.148i)7-s + (0.176 − 0.306i)8-s + (0.149 − 0.988i)9-s + (0.319 + 0.553i)10-s + (0.355 + 0.298i)11-s + (−0.467 + 0.176i)12-s + (0.319 + 1.81i)13-s + (0.0532 + 0.302i)14-s + (0.145 − 0.891i)15-s + (0.191 + 0.160i)16-s + (−0.761 − 1.31i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0208i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.72564 - 0.0180263i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.72564 - 0.0180263i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.347 - 1.96i)T \) |
| 3 | \( 1 + (-3.93 + 3.38i)T \) |
| good | 5 | \( 1 + (-7.73 + 6.49i)T + (21.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (-7.54 + 2.74i)T + (262. - 220. i)T^{2} \) |
| 11 | \( 1 + (-12.9 - 10.8i)T + (231. + 1.31e3i)T^{2} \) |
| 13 | \( 1 + (-14.9 - 84.8i)T + (-2.06e3 + 751. i)T^{2} \) |
| 17 | \( 1 + (53.3 + 92.4i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (17.8 - 30.9i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (85.4 + 31.0i)T + (9.32e3 + 7.82e3i)T^{2} \) |
| 29 | \( 1 + (16.6 - 94.5i)T + (-2.29e4 - 8.34e3i)T^{2} \) |
| 31 | \( 1 + (191. + 69.5i)T + (2.28e4 + 1.91e4i)T^{2} \) |
| 37 | \( 1 + (-79.7 - 138. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-43.2 - 245. i)T + (-6.47e4 + 2.35e4i)T^{2} \) |
| 43 | \( 1 + (-229. - 192. i)T + (1.38e4 + 7.82e4i)T^{2} \) |
| 47 | \( 1 + (-462. + 168. i)T + (7.95e4 - 6.67e4i)T^{2} \) |
| 53 | \( 1 + 317.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-127. + 107. i)T + (3.56e4 - 2.02e5i)T^{2} \) |
| 61 | \( 1 + (107. - 38.9i)T + (1.73e5 - 1.45e5i)T^{2} \) |
| 67 | \( 1 + (97.7 + 554. i)T + (-2.82e5 + 1.02e5i)T^{2} \) |
| 71 | \( 1 + (-124. - 216. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (479. - 830. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-222. + 1.26e3i)T + (-4.63e5 - 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-109. + 621. i)T + (-5.37e5 - 1.95e5i)T^{2} \) |
| 89 | \( 1 + (-427. + 741. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (241. + 202. i)T + (1.58e5 + 8.98e5i)T^{2} \) |
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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
−14.51438479956469086799726196505, −13.95207104983902336366726538297, −12.98024446892020363053793437362, −11.59687434955063544577938091345, −9.459645706424454555412820880517, −8.915839030017244182997739665140, −7.43900317513166148235492085102, −6.30822671391759510525453472061, −4.43488330106631077789300590476, −1.73034522114018109947171355217,
2.29349693014692758754109248019, 3.82234475557811352362338876949, 5.72009794605357317129276741471, 7.967076410227570494668572023747, 9.046323280868843034900883843595, 10.37842910290660942213745792093, 10.88292975628389753824381915259, 12.74313197975775918110487058472, 13.76150034719483999810280489268, 14.73807319432206091402402753450
