L(s) = 1 | + (0.623 − 0.781i)3-s + (−0.623 + 0.781i)5-s + (−0.222 − 0.974i)9-s + (0.222 − 0.974i)11-s + (0.222 − 0.974i)13-s + (0.222 + 0.974i)15-s + (0.900 − 0.433i)17-s + 19-s + (0.900 + 0.433i)23-s + (−0.222 − 0.974i)25-s + (−0.900 − 0.433i)27-s + (−0.900 + 0.433i)29-s + 31-s + (−0.623 − 0.781i)33-s + (−0.900 + 0.433i)37-s + ⋯ |
L(s) = 1 | + (0.623 − 0.781i)3-s + (−0.623 + 0.781i)5-s + (−0.222 − 0.974i)9-s + (0.222 − 0.974i)11-s + (0.222 − 0.974i)13-s + (0.222 + 0.974i)15-s + (0.900 − 0.433i)17-s + 19-s + (0.900 + 0.433i)23-s + (−0.222 − 0.974i)25-s + (−0.900 − 0.433i)27-s + (−0.900 + 0.433i)29-s + 31-s + (−0.623 − 0.781i)33-s + (−0.900 + 0.433i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.139204883 - 0.6177499204i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.139204883 - 0.6177499204i\) |
\(L(1)\) |
\(\approx\) |
\(1.140761409 - 0.3240909408i\) |
\(L(1)\) |
\(\approx\) |
\(1.140761409 - 0.3240909408i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.623 - 0.781i)T \) |
| 5 | \( 1 + (-0.623 + 0.781i)T \) |
| 11 | \( 1 + (0.222 - 0.974i)T \) |
| 13 | \( 1 + (0.222 - 0.974i)T \) |
| 17 | \( 1 + (0.900 - 0.433i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (0.900 + 0.433i)T \) |
| 29 | \( 1 + (-0.900 + 0.433i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (-0.900 + 0.433i)T \) |
| 41 | \( 1 + (-0.623 + 0.781i)T \) |
| 43 | \( 1 + (-0.623 - 0.781i)T \) |
| 47 | \( 1 + (-0.222 + 0.974i)T \) |
| 53 | \( 1 + (-0.900 - 0.433i)T \) |
| 59 | \( 1 + (0.623 + 0.781i)T \) |
| 61 | \( 1 + (0.900 - 0.433i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (0.900 + 0.433i)T \) |
| 73 | \( 1 + (0.222 + 0.974i)T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + (-0.222 - 0.974i)T \) |
| 89 | \( 1 + (0.222 + 0.974i)T \) |
| 97 | \( 1 - T \) |
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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
−27.05619557700063471060283556866, −26.31255630973323483647357772139, −25.279094881183435246845885640266, −24.4382619513088909055187187781, −23.282156549036364342163093086730, −22.40151821439710206116406110653, −21.06160064117971498041700777573, −20.639307710977433287273809046663, −19.61369349014655897460542845188, −18.8242607868757247335426197459, −17.16425082743674343455713219440, −16.4167480362273232817206072077, −15.48817009038667697606784334660, −14.64396694732388099180638802579, −13.55060267730917477801131327513, −12.33440460518324099084237671978, −11.37623883233559850133817792465, −9.986665323954010452431356711128, −9.174611228472744424051805681651, −8.20793778945674048176538786047, −7.13203917488521392125303362816, −5.275202394176214995525873434383, −4.36517050624453500345056252003, −3.39435667596992394604599919052, −1.697498561208820234200921020833,
1.08024334080341433465582389671, 3.004018843256696815075142157995, 3.430128377215506464036241252607, 5.50073853964195736523981085774, 6.75716753756006890042337559161, 7.68024551073316661925038902655, 8.49436588001092689185348676684, 9.85551776899075159135271535382, 11.19822429913628135180759027389, 12.01507895432693730947627738433, 13.26244294623397136005776192617, 14.12521528790590513601469868935, 15.01742711140099039971313841662, 15.99797799018921524507080552655, 17.41892513969925667857326600063, 18.54242614748955634819482596162, 18.99927177488565777007213081043, 20.00410670681670139709290928630, 20.91936006832704005692117404746, 22.300161697620524807590107191520, 23.10904617198153141631858827629, 24.05228399433815909207572351301, 24.96916764145765372810347611962, 25.824711700486934783492852381645, 26.82283943866104194413698373812