L(s) = 1 | + (−0.766 + 0.642i)3-s + (−0.173 − 0.984i)5-s + (0.173 − 0.984i)9-s + (0.5 − 0.866i)11-s + (0.173 − 0.984i)13-s + (0.766 + 0.642i)15-s + (−0.173 − 0.984i)17-s + (0.939 − 0.342i)23-s + (−0.939 + 0.342i)25-s + (0.5 + 0.866i)27-s + (0.939 − 0.342i)29-s − 31-s + (0.173 + 0.984i)33-s + (0.5 − 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯ |
L(s) = 1 | + (−0.766 + 0.642i)3-s + (−0.173 − 0.984i)5-s + (0.173 − 0.984i)9-s + (0.5 − 0.866i)11-s + (0.173 − 0.984i)13-s + (0.766 + 0.642i)15-s + (−0.173 − 0.984i)17-s + (0.939 − 0.342i)23-s + (−0.939 + 0.342i)25-s + (0.5 + 0.866i)27-s + (0.939 − 0.342i)29-s − 31-s + (0.173 + 0.984i)33-s + (0.5 − 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.809 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.809 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3182563678 - 0.9813899787i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3182563678 - 0.9813899787i\) |
\(L(1)\) |
\(\approx\) |
\(0.7789847190 - 0.2269930035i\) |
\(L(1)\) |
\(\approx\) |
\(0.7789847190 - 0.2269930035i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.766 + 0.642i)T \) |
| 5 | \( 1 + (-0.173 - 0.984i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.173 - 0.984i)T \) |
| 17 | \( 1 + (-0.173 - 0.984i)T \) |
| 23 | \( 1 + (0.939 - 0.342i)T \) |
| 29 | \( 1 + (0.939 - 0.342i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (0.5 - 0.866i)T \) |
| 41 | \( 1 + (0.173 + 0.984i)T \) |
| 43 | \( 1 + (-0.766 + 0.642i)T \) |
| 47 | \( 1 + (0.173 - 0.984i)T \) |
| 53 | \( 1 + (-0.173 + 0.984i)T \) |
| 59 | \( 1 + (-0.173 - 0.984i)T \) |
| 61 | \( 1 + (0.939 - 0.342i)T \) |
| 67 | \( 1 + (0.766 + 0.642i)T \) |
| 71 | \( 1 + (0.766 - 0.642i)T \) |
| 73 | \( 1 + (-0.766 + 0.642i)T \) |
| 79 | \( 1 + (-0.939 - 0.342i)T \) |
| 83 | \( 1 + (-0.5 + 0.866i)T \) |
| 89 | \( 1 + (0.766 + 0.642i)T \) |
| 97 | \( 1 + (-0.939 - 0.342i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−23.537625948825384829088283467708, −22.82917618103625400689873160077, −22.05814409115419679295565826048, −21.39755412500571407109918467950, −19.98363389565933916467493764697, −19.16736256701713250174738405541, −18.60012130677208560067972009690, −17.644067934131104773772451394212, −17.0883986412665730958429832945, −16.02268210285431096482650184631, −14.993351482396988135482892752, −14.23172401483666456379700473838, −13.22536075093547639551560364086, −12.28449669167711243860729955644, −11.50002486535867517907455234260, −10.784588200526490081233427129488, −9.876192551278776260087304602756, −8.60765645198468695088004021623, −7.34934250511576054754511921435, −6.820298836622097060301480178623, −6.05577660039478779685804906675, −4.78583403634886930651313595256, −3.7328268181105201044088699502, −2.29952769696700753048763165944, −1.37893537519913187232447199762,
0.34603689966718457679383838681, 1.07514460467544830876405931282, 3.02040654618698508866853416367, 4.07841505564354075640738470858, 5.04086013406815810671975415752, 5.702557963237108932553510439645, 6.7881521354285322294990009356, 8.13344900417067596014502462323, 9.01545957724611780680750492483, 9.78012469925966987206153677887, 10.94659502300499298110707260396, 11.56114515015733980452035754607, 12.52430921769811111383184195977, 13.28984941069588110633548319546, 14.50444671422070930744162606939, 15.55023671132508009125129683160, 16.21999730710525146354079507698, 16.84287069806783240541651532639, 17.66987099649680563240430601480, 18.56809916854874258728069670387, 19.833350082087758398252832869110, 20.44251998176800395646781427371, 21.34155262602209008651870656369, 21.99762715862884197096269723847, 23.059387565688801937270213979055