L(s) = 1 | + (−0.719 − 0.694i)2-s + (0.0348 + 0.999i)4-s + (−0.939 + 0.342i)5-s + (0.990 + 0.139i)7-s + (0.669 − 0.743i)8-s + (0.913 + 0.406i)10-s + (−0.990 − 0.139i)11-s + (0.241 + 0.970i)13-s + (−0.615 − 0.788i)14-s + (−0.997 + 0.0697i)16-s + (−0.309 − 0.951i)17-s + (−0.104 − 0.994i)19-s + (−0.374 − 0.927i)20-s + (0.615 + 0.788i)22-s + (0.615 + 0.788i)23-s + ⋯ |
L(s) = 1 | + (−0.719 − 0.694i)2-s + (0.0348 + 0.999i)4-s + (−0.939 + 0.342i)5-s + (0.990 + 0.139i)7-s + (0.669 − 0.743i)8-s + (0.913 + 0.406i)10-s + (−0.990 − 0.139i)11-s + (0.241 + 0.970i)13-s + (−0.615 − 0.788i)14-s + (−0.997 + 0.0697i)16-s + (−0.309 − 0.951i)17-s + (−0.104 − 0.994i)19-s + (−0.374 − 0.927i)20-s + (0.615 + 0.788i)22-s + (0.615 + 0.788i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.0009485038092 + 0.01034411990i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0009485038092 + 0.01034411990i\) |
\(L(1)\) |
\(\approx\) |
\(0.5956806464 - 0.09007855107i\) |
\(L(1)\) |
\(\approx\) |
\(0.5956806464 - 0.09007855107i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (-0.719 - 0.694i)T \) |
| 5 | \( 1 + (-0.939 + 0.342i)T \) |
| 7 | \( 1 + (0.990 + 0.139i)T \) |
| 11 | \( 1 + (-0.990 - 0.139i)T \) |
| 13 | \( 1 + (0.241 + 0.970i)T \) |
| 17 | \( 1 + (-0.309 - 0.951i)T \) |
| 19 | \( 1 + (-0.104 - 0.994i)T \) |
| 23 | \( 1 + (0.615 + 0.788i)T \) |
| 29 | \( 1 + (0.719 + 0.694i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (-0.997 - 0.0697i)T \) |
| 43 | \( 1 + (-0.961 + 0.275i)T \) |
| 47 | \( 1 + (0.559 + 0.829i)T \) |
| 53 | \( 1 + (0.978 + 0.207i)T \) |
| 59 | \( 1 + (-0.719 + 0.694i)T \) |
| 61 | \( 1 + (0.939 + 0.342i)T \) |
| 67 | \( 1 + (0.766 + 0.642i)T \) |
| 71 | \( 1 + (0.669 - 0.743i)T \) |
| 73 | \( 1 + (-0.309 + 0.951i)T \) |
| 79 | \( 1 + (-0.848 - 0.529i)T \) |
| 83 | \( 1 + (-0.961 + 0.275i)T \) |
| 89 | \( 1 + (0.978 - 0.207i)T \) |
| 97 | \( 1 + (-0.374 - 0.927i)T \) |
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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
−22.87123746740762106975941383225, −21.357602015569104819538581629844, −20.45614320606256175028755992047, −20.05012973793381875461614910823, −18.88256563214637164282578729623, −18.43957408857713430056517823568, −17.434068290840819514176806372549, −16.87912592428604379399982541572, −15.84002738750686719182586669700, −15.25852982904955628949068873261, −14.70519259184255007516505459164, −13.571208109618322042734050976173, −12.591158494995347400352551132716, −11.588096196350002500810245912249, −10.57821683235305955778981405691, −10.26956336948342304875649485267, −8.60715496265901796240782945317, −8.277900916349836534647294305531, −7.67534909393994254357761514374, −6.65159109380195332947498306886, −5.41665008143607554182253366943, −4.84809514944363934355249810305, −3.711358115120304134308518373412, −2.16795445435151854165724295351, −1.01669272556329246030908971773,
0.003622643497831929836234549124, 1.21293118822764901450924531695, 2.41278400123852270478514350768, 3.212606388311815333367552999023, 4.38387514514779842808459309030, 5.08482228142493189954238551424, 6.95013345061258535240074577258, 7.36287358583993446427095926660, 8.45710557225609045478514443691, 8.8827997138024337477539731502, 10.148545965515303429990735632262, 11.09622126693545507009583750778, 11.42430772829370666125643355389, 12.18796295060522015674879805989, 13.32255480583020217990408311803, 14.10431133570738760459652636057, 15.3809191683753443477538308095, 15.85280916689326698559107375308, 16.83995186241345711687257515417, 17.825653967897943238614218342380, 18.405411468071645868762775196, 19.052394590860381245689197578598, 19.89372017213529752843643710517, 20.644935121255867339139499633790, 21.37879121391209728351513761088