| L(s) = 1 | + (0.415 − 0.0467i)2-s + (2.68 − 1.68i)3-s + (−3.72 + 0.851i)4-s + (0.738 + 0.589i)5-s + (1.03 − 0.825i)6-s + (−0.577 + 2.53i)7-s + (−3.08 + 1.07i)8-s + (0.449 − 0.932i)9-s + (0.334 + 0.209i)10-s + (−8.65 − 3.02i)11-s + (−8.56 + 8.56i)12-s + (−4.51 − 9.37i)13-s + (−0.121 + 1.07i)14-s + (2.97 + 0.335i)15-s + (12.5 − 6.04i)16-s + (21.4 + 21.4i)17-s + ⋯ |
| L(s) = 1 | + (0.207 − 0.0233i)2-s + (0.894 − 0.561i)3-s + (−0.932 + 0.212i)4-s + (0.147 + 0.117i)5-s + (0.172 − 0.137i)6-s + (−0.0824 + 0.361i)7-s + (−0.385 + 0.134i)8-s + (0.0499 − 0.103i)9-s + (0.0334 + 0.0209i)10-s + (−0.786 − 0.275i)11-s + (−0.714 + 0.714i)12-s + (−0.347 − 0.720i)13-s + (−0.00866 + 0.0769i)14-s + (0.198 + 0.0223i)15-s + (0.784 − 0.377i)16-s + (1.26 + 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.225i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.11334 - 0.127015i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11334 - 0.127015i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (24.3 - 15.7i)T \) |
| good | 2 | \( 1 + (-0.415 + 0.0467i)T + (3.89 - 0.890i)T^{2} \) |
| 3 | \( 1 + (-2.68 + 1.68i)T + (3.90 - 8.10i)T^{2} \) |
| 5 | \( 1 + (-0.738 - 0.589i)T + (5.56 + 24.3i)T^{2} \) |
| 7 | \( 1 + (0.577 - 2.53i)T + (-44.1 - 21.2i)T^{2} \) |
| 11 | \( 1 + (8.65 + 3.02i)T + (94.6 + 75.4i)T^{2} \) |
| 13 | \( 1 + (4.51 + 9.37i)T + (-105. + 132. i)T^{2} \) |
| 17 | \( 1 + (-21.4 - 21.4i)T + 289iT^{2} \) |
| 19 | \( 1 + (-14.2 + 22.6i)T + (-156. - 325. i)T^{2} \) |
| 23 | \( 1 + (2.27 + 2.85i)T + (-117. + 515. i)T^{2} \) |
| 31 | \( 1 + (-21.7 + 2.44i)T + (936. - 213. i)T^{2} \) |
| 37 | \( 1 + (46.0 - 16.1i)T + (1.07e3 - 853. i)T^{2} \) |
| 41 | \( 1 + (25.3 - 25.3i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 + (-2.78 + 24.6i)T + (-1.80e3 - 411. i)T^{2} \) |
| 47 | \( 1 + (-7.49 + 21.4i)T + (-1.72e3 - 1.37e3i)T^{2} \) |
| 53 | \( 1 + (18.6 - 23.3i)T + (-625. - 2.73e3i)T^{2} \) |
| 59 | \( 1 - 18.2T + 3.48e3T^{2} \) |
| 61 | \( 1 + (78.8 - 49.5i)T + (1.61e3 - 3.35e3i)T^{2} \) |
| 67 | \( 1 + (-24.5 + 50.9i)T + (-2.79e3 - 3.50e3i)T^{2} \) |
| 71 | \( 1 + (-56.1 - 116. i)T + (-3.14e3 + 3.94e3i)T^{2} \) |
| 73 | \( 1 + (-47.2 - 5.32i)T + (5.19e3 + 1.18e3i)T^{2} \) |
| 79 | \( 1 + (13.7 + 39.2i)T + (-4.87e3 + 3.89e3i)T^{2} \) |
| 83 | \( 1 + (29.4 + 129. i)T + (-6.20e3 + 2.98e3i)T^{2} \) |
| 89 | \( 1 + (-32.3 + 3.64i)T + (7.72e3 - 1.76e3i)T^{2} \) |
| 97 | \( 1 + (36.1 + 22.6i)T + (4.08e3 + 8.47e3i)T^{2} \) |
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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
−17.08101444425267158682914534700, −15.31676900216299649223155591864, −14.17186740219259830427231043112, −13.30491528969641525440540663359, −12.35758688987759699301752298671, −10.22295439552180853806801587786, −8.697173760857372190145326127897, −7.76026180131083907428655894018, −5.39705689307691073367395271469, −3.05971723940551040384636244451,
3.58633049243738481376447072993, 5.25931041055721269380504203799, 7.78739878474684240259086132770, 9.341517216760252394283863023683, 9.984204537705855650000623892272, 12.17285986553491679311606352355, 13.73195394915243870995370153468, 14.27845399368517752772454627499, 15.50257035069779237635652054229, 16.87732382497038192593113041750
