L(s) = 1 | − 2-s − 2.90·3-s + 4-s − 1.99·5-s + 2.90·6-s − 0.299·7-s − 8-s + 5.42·9-s + 1.99·10-s − 4.57·11-s − 2.90·12-s + 2.85·13-s + 0.299·14-s + 5.79·15-s + 16-s + 2.21·17-s − 5.42·18-s + 5.38·19-s − 1.99·20-s + 0.868·21-s + 4.57·22-s + 23-s + 2.90·24-s − 1.01·25-s − 2.85·26-s − 7.04·27-s − 0.299·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.67·3-s + 0.5·4-s − 0.892·5-s + 1.18·6-s − 0.113·7-s − 0.353·8-s + 1.80·9-s + 0.631·10-s − 1.37·11-s − 0.838·12-s + 0.792·13-s + 0.0799·14-s + 1.49·15-s + 0.250·16-s + 0.536·17-s − 1.27·18-s + 1.23·19-s − 0.446·20-s + 0.189·21-s + 0.974·22-s + 0.208·23-s + 0.592·24-s − 0.202·25-s − 0.560·26-s − 1.35·27-s − 0.0565·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4180111934\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4180111934\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
good | 3 | \( 1 + 2.90T + 3T^{2} \) |
| 5 | \( 1 + 1.99T + 5T^{2} \) |
| 7 | \( 1 + 0.299T + 7T^{2} \) |
| 11 | \( 1 + 4.57T + 11T^{2} \) |
| 13 | \( 1 - 2.85T + 13T^{2} \) |
| 17 | \( 1 - 2.21T + 17T^{2} \) |
| 19 | \( 1 - 5.38T + 19T^{2} \) |
| 29 | \( 1 + 5.80T + 29T^{2} \) |
| 31 | \( 1 - 7.60T + 31T^{2} \) |
| 37 | \( 1 + 1.64T + 37T^{2} \) |
| 41 | \( 1 - 0.356T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 + 1.51T + 47T^{2} \) |
| 53 | \( 1 - 8.03T + 53T^{2} \) |
| 59 | \( 1 - 5.74T + 59T^{2} \) |
| 61 | \( 1 - 8.87T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 + 6.00T + 71T^{2} \) |
| 73 | \( 1 - 0.198T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.83038966517540621456578788677, −7.46624174596319090869497845582, −6.74836770831111970068060733829, −5.76681158262099762819354494785, −5.53632948114708198190203582060, −4.60516389097259000901204461088, −3.71589425770697030947725581557, −2.74036555268833301173319214069, −1.29105430138844673537535912360, −0.46388110636291734743027863837,
0.46388110636291734743027863837, 1.29105430138844673537535912360, 2.74036555268833301173319214069, 3.71589425770697030947725581557, 4.60516389097259000901204461088, 5.53632948114708198190203582060, 5.76681158262099762819354494785, 6.74836770831111970068060733829, 7.46624174596319090869497845582, 7.83038966517540621456578788677