L(s) = 1 | + (0.237 + 0.971i)2-s + (−0.887 + 0.460i)4-s + (0.119 − 0.992i)5-s + (−0.657 − 0.753i)8-s + (0.992 − 0.119i)10-s + (0.427 − 0.904i)11-s + (0.0224 − 0.999i)13-s + (0.575 − 0.817i)16-s + (0.215 − 0.976i)17-s + (0.998 + 0.0523i)19-s + (0.351 + 0.936i)20-s + (0.979 + 0.200i)22-s + (−0.772 + 0.635i)23-s + (−0.971 − 0.237i)25-s + (0.976 − 0.215i)26-s + ⋯ |
L(s) = 1 | + (0.237 + 0.971i)2-s + (−0.887 + 0.460i)4-s + (0.119 − 0.992i)5-s + (−0.657 − 0.753i)8-s + (0.992 − 0.119i)10-s + (0.427 − 0.904i)11-s + (0.0224 − 0.999i)13-s + (0.575 − 0.817i)16-s + (0.215 − 0.976i)17-s + (0.998 + 0.0523i)19-s + (0.351 + 0.936i)20-s + (0.979 + 0.200i)22-s + (−0.772 + 0.635i)23-s + (−0.971 − 0.237i)25-s + (0.976 − 0.215i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.826609064 - 0.6536629467i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.826609064 - 0.6536629467i\) |
\(L(1)\) |
\(\approx\) |
\(1.176136316 + 0.1199718749i\) |
\(L(1)\) |
\(\approx\) |
\(1.176136316 + 0.1199718749i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + (0.237 + 0.971i)T \) |
| 5 | \( 1 + (0.119 - 0.992i)T \) |
| 11 | \( 1 + (0.427 - 0.904i)T \) |
| 13 | \( 1 + (0.0224 - 0.999i)T \) |
| 17 | \( 1 + (0.215 - 0.976i)T \) |
| 19 | \( 1 + (0.998 + 0.0523i)T \) |
| 23 | \( 1 + (-0.772 + 0.635i)T \) |
| 29 | \( 1 + (0.493 + 0.869i)T \) |
| 31 | \( 1 + (0.104 - 0.994i)T \) |
| 37 | \( 1 + (0.712 - 0.701i)T \) |
| 43 | \( 1 + (-0.512 - 0.858i)T \) |
| 47 | \( 1 + (-0.519 + 0.854i)T \) |
| 53 | \( 1 + (0.953 + 0.301i)T \) |
| 59 | \( 1 + (0.873 + 0.486i)T \) |
| 61 | \( 1 + (0.986 + 0.163i)T \) |
| 67 | \( 1 + (0.933 - 0.358i)T \) |
| 71 | \( 1 + (0.869 + 0.493i)T \) |
| 73 | \( 1 + (0.997 - 0.0747i)T \) |
| 79 | \( 1 + (0.965 - 0.258i)T \) |
| 83 | \( 1 + (0.900 + 0.433i)T \) |
| 89 | \( 1 + (-0.480 + 0.877i)T \) |
| 97 | \( 1 + (0.156 - 0.987i)T \) |
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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
−17.94748983829794360032486264632, −17.4189216129149972524364025993, −16.57332843365367454355264632559, −15.6417828527440514484991465308, −14.807856718013479651057652972100, −14.49183131976206863660203937701, −13.83942109792020640427193272017, −13.215147591242253055156257030097, −12.30569373424287580185271206079, −11.806045266684346091771443317360, −11.3057401816909716630873027466, −10.45386787847786089751500299959, −9.86468388362989443035242018035, −9.562534681069650410485629509890, −8.48736170758275996899483255442, −7.865036742098504722508159397514, −6.69932615948321959168764736187, −6.46526499633828069301366639002, −5.42986654004803764675218906235, −4.62632661350962286872682292786, −3.895287881611223062846096249175, −3.380641069040560204772498964041, −2.34199274046142371330498129887, −1.965544098284483722181870024544, −1.021978533291026184633593410736,
0.58373915012617292570028157145, 1.055900774299823005024423922143, 2.48990263421204929317078038827, 3.46742029265093801302469795337, 3.955473802034298064135738116445, 5.01125022510282663190653925048, 5.431644022812067065740169778101, 5.933800079606689479380921150084, 6.84912986676247604004500169829, 7.709994118901141436057236136310, 8.12541731405399108486618257175, 8.84795801162719204436634766469, 9.517027765817122484699895924033, 9.97740222350928931842320711639, 11.20286823909646991574765116873, 11.911566437812181362019892266835, 12.49545657902653486818316280095, 13.25362500213870708794576566817, 13.77184658456915470780893235508, 14.229354723642588850439119244, 15.17756828185009836965085702705, 15.81427663110345027576903795364, 16.358510924273403917939949633434, 16.69902193054635942493040231159, 17.63502709780778031173590024638