L(s) = 1 | + (−0.362 − 0.931i)2-s + (0.309 − 0.951i)3-s + (−0.736 + 0.676i)4-s + (0.696 + 0.717i)5-s + (−0.998 + 0.0570i)6-s + (0.516 − 0.856i)7-s + (0.897 + 0.441i)8-s + (−0.809 − 0.587i)9-s + (0.415 − 0.909i)10-s + (0.415 + 0.909i)12-s + (0.198 − 0.980i)13-s + (−0.985 − 0.170i)14-s + (0.897 − 0.441i)15-s + (0.0855 − 0.996i)16-s + (0.610 − 0.791i)17-s + (−0.254 + 0.967i)18-s + ⋯ |
L(s) = 1 | + (−0.362 − 0.931i)2-s + (0.309 − 0.951i)3-s + (−0.736 + 0.676i)4-s + (0.696 + 0.717i)5-s + (−0.998 + 0.0570i)6-s + (0.516 − 0.856i)7-s + (0.897 + 0.441i)8-s + (−0.809 − 0.587i)9-s + (0.415 − 0.909i)10-s + (0.415 + 0.909i)12-s + (0.198 − 0.980i)13-s + (−0.985 − 0.170i)14-s + (0.897 − 0.441i)15-s + (0.0855 − 0.996i)16-s + (0.610 − 0.791i)17-s + (−0.254 + 0.967i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5320377641 - 0.8720137048i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5320377641 - 0.8720137048i\) |
\(L(1)\) |
\(\approx\) |
\(0.7720707219 - 0.6583728028i\) |
\(L(1)\) |
\(\approx\) |
\(0.7720707219 - 0.6583728028i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
good | 2 | \( 1 + (-0.362 - 0.931i)T \) |
| 3 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (0.696 + 0.717i)T \) |
| 7 | \( 1 + (0.516 - 0.856i)T \) |
| 13 | \( 1 + (0.198 - 0.980i)T \) |
| 17 | \( 1 + (0.610 - 0.791i)T \) |
| 19 | \( 1 + (-0.564 + 0.825i)T \) |
| 23 | \( 1 + (-0.654 + 0.755i)T \) |
| 29 | \( 1 + (-0.0285 - 0.999i)T \) |
| 31 | \( 1 + (-0.870 + 0.491i)T \) |
| 37 | \( 1 + (0.993 + 0.113i)T \) |
| 41 | \( 1 + (0.774 - 0.633i)T \) |
| 43 | \( 1 + (-0.142 + 0.989i)T \) |
| 47 | \( 1 + (-0.254 - 0.967i)T \) |
| 53 | \( 1 + (0.0855 + 0.996i)T \) |
| 59 | \( 1 + (0.774 + 0.633i)T \) |
| 61 | \( 1 + (-0.362 + 0.931i)T \) |
| 67 | \( 1 + (0.841 + 0.540i)T \) |
| 71 | \( 1 + (0.941 - 0.336i)T \) |
| 73 | \( 1 + (-0.921 - 0.389i)T \) |
| 79 | \( 1 + (-0.466 + 0.884i)T \) |
| 83 | \( 1 + (0.974 + 0.226i)T \) |
| 89 | \( 1 + (-0.959 + 0.281i)T \) |
| 97 | \( 1 + (0.696 - 0.717i)T \) |
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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
−28.756687920057367214572422907870, −28.129303055684681903490117790127, −27.422788172208493786336125208526, −26.05218240245201537535251065030, −25.60777227817924951188561204300, −24.48850274832985440544759233467, −23.64792336061669906473628067756, −21.982723118127378462910423555027, −21.5019774782190821513729077748, −20.25049810338429783014274839452, −18.96272597231117020799407498721, −17.769518043845082368650610425, −16.74577368966082793512260241636, −16.04690278694232386869199723480, −14.8214560895265019593728384819, −14.18377016079763815560118597409, −12.79482744770884841429871503468, −11.02701471215528526383171595694, −9.70522288095534383301895069364, −8.92965198790989078200044411897, −8.18457756572871853901481677206, −6.22788019389022595320564935742, −5.23115554051795353128545639154, −4.26720350519793642238695472940, −1.97906036497666934846854949818,
1.22268804569264327917930213694, 2.4661855985307626224373298473, 3.68134534999211637244183292511, 5.68992038807700607850903240778, 7.32676443811782997772346165735, 8.09545961276603164313292591018, 9.63572060913863430535356928609, 10.63294700549299763502405585074, 11.698153519260574096664590416279, 12.986530963712354741746654466550, 13.79449623003394225345668814394, 14.611380063234435225300956911336, 16.85465171607879881512912034785, 17.84083482277370646408069424322, 18.338651238169629214932054222949, 19.49709785805555901411876698895, 20.4378845697532586575566926697, 21.28863112466341320677616154903, 22.71982440013276580042666659034, 23.36365330469535438573426822572, 24.98475249610403432471855391818, 25.75716834310287216273472811758, 26.75881503979359631600599515248, 27.74529419590623920492472588474, 29.1774005554031640034247700593