| L(s) = 1 | + (−0.160 + 0.701i)2-s + (1.15 − 0.558i)3-s + (6.74 + 3.24i)4-s + (−0.752 + 3.29i)5-s + (0.206 + 0.903i)6-s + (19.5 − 9.43i)7-s + (−6.95 + 8.71i)8-s + (−15.8 + 19.8i)9-s + (−2.19 − 1.05i)10-s + (−31.6 − 39.6i)11-s + 9.62·12-s + (−40.9 − 51.3i)13-s + (3.48 + 15.2i)14-s + (0.967 + 4.24i)15-s + (32.3 + 40.5i)16-s − 3.51·17-s + ⋯ |
| L(s) = 1 | + (−0.0566 + 0.248i)2-s + (0.223 − 0.107i)3-s + (0.842 + 0.405i)4-s + (−0.0673 + 0.294i)5-s + (0.0140 + 0.0614i)6-s + (1.05 − 0.509i)7-s + (−0.307 + 0.385i)8-s + (−0.585 + 0.733i)9-s + (−0.0693 − 0.0334i)10-s + (−0.867 − 1.08i)11-s + 0.231·12-s + (−0.874 − 1.09i)13-s + (0.0665 + 0.291i)14-s + (0.0166 + 0.0729i)15-s + (0.504 + 0.633i)16-s − 0.0501·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.36093 + 0.311279i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.36093 + 0.311279i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (34.2 + 152. i)T \) |
| good | 2 | \( 1 + (0.160 - 0.701i)T + (-7.20 - 3.47i)T^{2} \) |
| 3 | \( 1 + (-1.15 + 0.558i)T + (16.8 - 21.1i)T^{2} \) |
| 5 | \( 1 + (0.752 - 3.29i)T + (-112. - 54.2i)T^{2} \) |
| 7 | \( 1 + (-19.5 + 9.43i)T + (213. - 268. i)T^{2} \) |
| 11 | \( 1 + (31.6 + 39.6i)T + (-296. + 1.29e3i)T^{2} \) |
| 13 | \( 1 + (40.9 + 51.3i)T + (-488. + 2.14e3i)T^{2} \) |
| 17 | \( 1 + 3.51T + 4.91e3T^{2} \) |
| 19 | \( 1 + (22.9 + 11.0i)T + (4.27e3 + 5.36e3i)T^{2} \) |
| 23 | \( 1 + (-21.3 - 93.7i)T + (-1.09e4 + 5.27e3i)T^{2} \) |
| 31 | \( 1 + (-6.98 + 30.5i)T + (-2.68e4 - 1.29e4i)T^{2} \) |
| 37 | \( 1 + (-113. + 141. i)T + (-1.12e4 - 4.93e4i)T^{2} \) |
| 41 | \( 1 + 287.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (-44.7 - 196. i)T + (-7.16e4 + 3.44e4i)T^{2} \) |
| 47 | \( 1 + (-199. - 250. i)T + (-2.31e4 + 1.01e5i)T^{2} \) |
| 53 | \( 1 + (97.5 - 427. i)T + (-1.34e5 - 6.45e4i)T^{2} \) |
| 59 | \( 1 - 433.T + 2.05e5T^{2} \) |
| 61 | \( 1 + (-683. + 329. i)T + (1.41e5 - 1.77e5i)T^{2} \) |
| 67 | \( 1 + (129. - 161. i)T + (-6.69e4 - 2.93e5i)T^{2} \) |
| 71 | \( 1 + (286. + 359. i)T + (-7.96e4 + 3.48e5i)T^{2} \) |
| 73 | \( 1 + (8.19 + 35.9i)T + (-3.50e5 + 1.68e5i)T^{2} \) |
| 79 | \( 1 + (796. - 998. i)T + (-1.09e5 - 4.80e5i)T^{2} \) |
| 83 | \( 1 + (-932. - 448. i)T + (3.56e5 + 4.47e5i)T^{2} \) |
| 89 | \( 1 + (-194. + 851. i)T + (-6.35e5 - 3.05e5i)T^{2} \) |
| 97 | \( 1 + (1.34e3 + 649. i)T + (5.69e5 + 7.13e5i)T^{2} \) |
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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
−16.79375149808040935235602631557, −15.48082893016559876217238421476, −14.42237283725994961732889365521, −13.11714741619053328731189296960, −11.37073720681462049709566910192, −10.68810315956992299402923852044, −8.194927857888342709549612552316, −7.54054029617442417597425879904, −5.47333903763756715186589920797, −2.76160371000943309261986641210,
2.24519109221282804094619061197, 5.00900682237850006194521918641, 6.94252883665174497060690647591, 8.664067575893256945306846091333, 10.15114448423929806299741904174, 11.59245248993409763516442759620, 12.41242328764923980122831347415, 14.60536630894024445602231773814, 15.01021541606379995970085713964, 16.47509115771964927088575416008
